Device, method and system of pricing financial instruments

ABSTRACT

Some demonstrative embodiments include methods, devices and systems of pricing financial instruments. In one embodiment, a pricing module may be configured to receive first input data corresponding to at least one parameter defining a first option on an underlying asset and second input data corresponding to at least one current market condition relating to said underlying asset, and, based on said first and second input data, to determine a price of the first option according to a volatility smile satisfying a first criterion relating to a sum of a first correction corresponding to the first option and a second correction corresponding to a second option representing a position opposite to a position of a the first option and having substantially a same absolute delta value as the first option, wherein the first correction relates to a difference between a theoretical price of the first option and the price of the first option according to the volatility smile, and wherein the second correction relates to a difference between a theoretical price of the second option and the price of the second option according to the volatility smile. Other embodiments are described and claimed.

CROSS REFERENCE

This application claims the benefit of and priority from U.S.Provisional Patent application 61/291,942, entitled “Method and systemof pricing financial instruments”, filed Jan. 4, 2010, the entiredisclosure of which is incorporated herein by reference.

FIELD

The disclosure relates generally to financial instruments and, morespecifically, to methods and systems for pricing, e.g., real-timepricing, of options and/or for providing automatic trading capabilities.

BACKGROUND

Pricing financial instruments is a complex art requiring substantialexpertise and experience. Trading financial instruments, such asoptions, involves a sophisticated process of pricing typically performedby a trader.

The term “option” in the context of the present application is broadlydefined as any financial instrument having option-like properties, e.g.,any financial derivative including an option or an option-likecomponent. This category of financial instruments may include any typeof option or option-like financial instrument, relating to someunderlying asset. Assets as used in this application include anything ofvalue; tangible or non-tangible, financial or non-financial, forexample, stocks; currencies; commodities, e.g., oil, metals, or sugar;interest rates; forward-rate agreements (FRA); swaps; futures; bonds;weather, e.g., the temperature at a certain area; electricity; gasemission; credit; mortgages; indices; and the like. For example, as usedherein, options range from a simple Vanilla option on a single stock andup to complex convertible bonds whose convertibility depends on somekey, e.g., the weather.

The term “Exchange” in the context of the present application relates toany one or more exchanges throughout the world, and includes allassets/securities, which may be traded in these exchanges. The terms“submit a price to the exchange”, “submit a quote to the exchange”, andthe like generally refer to actions that a trader may perform to submita bid and/or offer prices for trading in the exchange. The price may betransferred from the trader to the exchange, for example, by a broker,by online trading, on a special communication network, through aclearing house system, and/or using in any other desired system and/ormethod.

The price of an asset for immediate, e.g., 1 or 2 business days,delivery is called the spot price. For an asset sold in an optioncontract, the strike price is the agreed upon price at which the deal isexecuted if the option is exercised. For example, a stock optioninvolves buying or selling a stock. The spot price is the current stockprice on the exchange in which is the stock is traded. The strike priceis the agreed upon price to buy/sell the stock if the option isexercised.

To facilitate trading of options and other financial instruments, amarket maker suggests a bid price and offer price (also called askprice) for a certain option. The bid price is the price at which themarket maker is willing to purchase the option and the offer price isthe price at which the market maker is willing to sell the option. As amarket practice, a first trader interested in a certain option may ask asecond trader for a quote, e.g., without indicating whether the firsttrader is interested to buy or to sell the option. The second traderquotes both the bid and offer prices, not knowing whether the firsttrader is interested in selling or buying the option. The market makermay earn a margin by buying options at a first price and selling them ata second price, e.g., higher than the first price. The differencebetween the offer and bid prices is referred to as bid-offer spread.

A call option is the right to buy an asset at a certain price (“thestrike”) at a certain time, e.g., on a certain date. A put option is theright to sell an asset at a strike price at a certain time, e.g., on acertain date. Every option has an expiration time in which the optionceases to exist. Prior to the option expiration time, the holder of theoption may determine whether or not to exercise the option, depending onthe prevailing spot price for the underlying asset. If the spot price atexpiration is lower than the strike price, the holder will choose not toexercise the call option and lose only the cost of the option itself.However, if the strike is lower than the spot, the holder of the calloption will exercise the right to buy the underlying asset at the strikeprice making a profit equal to the difference between the spot and thestrike prices. The cost of the option is also referred to as thepremium.

A forward rate is defined as the predetermined rate or price of anasset, at which an agreed upon future transaction will take place. Theforward rate may be calculated based on a current rate of the asset, acurrent interest rate prevailing in the market, expected dividends (forstocks), cost of carry (for commodities), and/or other parametersdepending on the underlying asset of the option.

An at-the-money forward option (ATM) is an option whose strike is equalto the forward rate of the asset. In some fields, the at-the-moneyforward options are generically referred to as at-the-money options, asis the common terminology in the commodities and interest rates options.The at the money equity options are actually the at the money spot, i.e.where the strike is the current spot rate or price.

An in-the-money call option is a call option whose strike is below theforward rate of the underlying asset, and an in the-money put option isa put option whose strike is above the forward rate of the underlyingasset. An out-of-the-money call option is a call option whose strike isabove the forward rate of the underlying asset, and an out-of-the-moneyput option is a put option whose strike is below the forward rate of theunderlying asset.

An exotic option, in the context of this application, is a generic namereferring to any type of option other than a standard Vanilla option.While certain types of exotic options have been extensively andfrequently traded over the years, and are still traded today, othertypes of exotic options had been used in the past but are no longer inuse today. Currently, the most common exotic options include “barrier”options, “digital” options, “binary” options, “partial barrier” options(also known as “window” options), “average” options, “compound” optionsand “quanto” options. Some exotic options can be described as a complexversion of the standard (Vanilla) option. For example, barrier optionsare exotic options where the payoff depends on whether the underlyingasset's price reaches a certain level, hereinafter referred to as“trigger”, during a certain period of time. The “pay off” of an optionis defined as the cash realized by the holder of the option upon itsexpiration. There are generally two types of barrier options, namely, aknock-out option and a knock-in option. A knock-out option is an optionthat terminates if and when the spot reaches the trigger. A knock-inoption comes into existence only when the underlying asset's pricereaches the trigger. It is noted that the combined effect of a knock-outoption with strike K and trigger B and a knock-in option with strike Kand trigger B, both having the same expiration, is equivalent to acorresponding Vanilla option with strike K. Thus, knock-in options canbe priced by pricing corresponding knock-out and vanilla options.Similarly, a one-touch option can be decomposed into two knock-in calloptions and two knock-in put options, a double no-touch option can bedecomposed into two double knock-out options, and so on. It isappreciated that there are many other types of exotic options known inthe art.

Certain types of options, e.g., Vanilla options, are commonlycategorized as either European or American. A European option can beexercised only upon its expiration. An American option can be exercisedat any time after purchase and before expiration. For example, anAmerican Vanilla option has all the properties of the Vanilla optiontype described above, with the additional property that the owner canexercise the option at any time up to and including the option'sexpiration date. As is known in the art, the right to exercise anAmerican option prior to expiration makes American options moreexpensive than corresponding European options.

Generally in this application, the term “Vanilla” refers to a Europeanstyle Vanilla option. European Vanilla options are the most commonlytraded options; they are typically traded over the counter (OTC).American Vanilla options are more popular in the exchanges and, ingeneral, are more difficult to price.

U.S. Pat. No. 5,557,517 (“the '517 patent”) describes a method ofpricing American Vanilla options for trading in a certain exchange. Thispatent describes a method of pricing Call and Put American Vanillaoptions, where the price of the option depends on a constant margin orcommission required by the market maker.

The method of the '517 patent ignores data that may affect the price ofthe option, except for the current price of the underlying asset and,thus, this method can lead to serious errors, for example, an absurdresult of a negative option price. Clearly, this method does not emulatethe way American style Vanilla options are priced in real markets.

The Black-Scholes (BS) model (developed in 1973) is a widely acceptedmethod for valuing options. This model calculates a theoretical value(TV) for options based on the probability of the payout, which iscommonly used as a starting point for approximating option prices. Thismodel is based on a presumption that the change in the spot price of theasset generally follows a Brownian motion, as is known in the art. Usingsuch Brownian motion model, known also as a stochastic process, one maycalculate the theoretical price of any type of financial derivative,either analytically or numerically. For example, it is common tocalculate the theoretical price of complicated financial derivativesthrough simulation techniques, such as the Monte-Carlo method,introduced by Boyle in 1977. Such techniques may be useful incalculating the theoretical value of an option, provided that a computerbeing used is sufficiently powerful to handle all the calculationsinvolved. In the simulation method, the computer generates manypropagation paths for the underlying asset, starting at the trade timeand ending at the time of the option expiry. Each path is discrete andgenerally follows the Brownian motion probability, but may be generatedas densely as necessary by reducing the time lapse between each move ofthe underlying asset. Thus, if the option is path-dependant, each pathis followed and only the paths that satisfy the conditions of the optionare taken into account. The end results of each such path are summarizedand lead to the theoretical price of the derivative.

The original Black-Scholes model was derived for calculating theoreticalprices of European Vanilla options, where the price of the option isdescribed by a relatively simple formula. However, it should beunderstood that any reference in this application to the Black-Scholesmodel refers to use of the Black-Scholes model or any other suitablemodel for evaluating the behavior of the underlying asset, e.g.,assuming a stochastic process (Brownian motion), and/or for evaluatingthe price of any type of option, including exotic options. Furthermore,this application is general and independent of the way in which thetheoretical value of the option is obtained. It can be derivedanalytically, numerically, using any kind of simulation method or anyother technique available.

For example, U.S. Pat. No. 6,061,662 (“the '662 patent”) describes amethod of evaluating the theoretical price of an option using aMonte-Carlo method based on historical data. The simulation method ofthe '662 patent uses stochastic historical data with a predetermineddistribution function in order to evaluate the theoretical price ofoptions. Examples is the '662 patent are used to illustrate that thismethod generates results which are very similar to those obtained byapplying the Black-Scholes model to Vanilla options. Unfortunately,methods based on historical data alone are not relevant for simulatingfinancial markets, even for the purpose of theoretical valuation. Forexample, one of the most important parameters used for valuation ofoptions is the volatility of the underlying asset, which is a measurefor how the price and/or rate of the underlying asset may fluctuate. Itis well known that the financial markets use a predicted, or anexpected, value for the volatility of the underlying assets, which oftendeviates dramatically from the historical data. In market terms,expected volatility is often referred to as “implied volatility”, and isdifferentiated from “historical volatility”. For example, the impliedvolatility tends to be much higher than the historical volatility of theunderlying asset before a major event, such as risk of war, and inanticipation of or during a financial crisis.

It is appreciated by persons skilled in the art that the Black-Scholesmodel is a limited approximation that may yield results very far fromreal market prices and, thus, corrections to the Black-Scholes modelmust generally be added by traders. For example, in the Foreign Exchange(FX) Vanilla market, and in commodities, the market trades in volatilityterms and the translation to option price is performed through use ofthe Black-Scholes formula. In fact, traders commonly refer to using theBlack-Scholes model as “using the wrong volatility with the wrong modelto get the right price”.

In order to adjust the BS price, in the Vanilla market, traders usedifferent volatilities for different strikes, i.e., instead of using onevolatility per asset per expiration date, as is required by the BSmodel, a trader may use different volatility values for a given assetdepending on the strike price. This adjustment is known as volatility“smile” adjustment. The origin of the term “smile”, in this context, isthe typical shape of the volatility vs. strike, which is similar to aflat “U” shape (smile).

The phrase “market price of an option” is used herein to distinguishbetween the single value produced by some known models, such as theBlack-Scholes model, and the actual bid and offer prices traded in thereal market. For example, for some options, the market bid side may betwice the Black-Scholes model price and the offer side may be threetimes the Black-Scholes model price.

Many exotic options are characterized by discontinuity of the payoutand, therefore, a discontinuity in some of the risk parameters near thetrigger(s). This discontinuity prevents an oversimplified model such asthe Black-Scholes model from taking into account the difficulty inrisk-managing the option. Furthermore, due to the peculiar profile ofsome exotic options, there may be significant transaction costsassociated with re-hedging some of the risk factors. Existing models,such as the Black-Scholes model, completely ignore such risk factors.

Several options pricing models were introduced since 1973, but none ofthese models was able to replicate the market prices universally and/orconsistently. The most famous pricing models include, the stochasticvolatility model, which assumes that the volatility itself is anotherstochastic process correlated with the underlying process; the localvolatility model, where the volatility is a function of time and theunderlying asset; and Libor based models, such as BGM, which generatethe swaption prices from the Libor rates which are correlated stochasticprocesses.

Many factors may be taken into account in calculating option prices andcorrections. The term “Factor” is used herein broadly as anyquantifiable or computable value relating to the subject option. Some ofthe notable factors are defined as follows.

Volatility (“Vol”) is a measure of the fluctuation of the returnrealized on an asset, e.g., a daily return. An indication of the orderof magnitude the volatility can be obtained by historical volatility,i.e., the standard deviation of the daily return of the assets for acertain past period.

However, the markets trade based on a volatility that reflects themarket expectations of the standard deviation in the future. Thevolatility reflecting market expectations is called implied volatility.In order to buy/sell volatility one commonly trades Vanilla options. Forexample, in the foreign exchange market, the implied volatilities of ATMVanilla options for frequently used option dates and currency pairs areavailable to users in real-time, e.g., via screens such as REUTERS,Bloomberg or directly from FX option brokers.

Volatility smile, as discussed above, relates to the behavior of theimplied volatility with respect to the strike, i.e., the impliedvolatility as a function of the strike, where the implied volatility forthe ATM strike is the given ATM volatility in the market. Typically theplot of the implied volatility as a function of the strike shows aminimum that looks like a smile. For example, usually in equity optionsthe minimum volatility is below the ATM strike.

Delta is the rate of change in the price of an option in response tochanges in the price of the underlying asset; in other words, it is apartial derivative of the option price with respect to the spot. Forexample, a 25 delta call option is defined as follows: if against buyingthe option on one unit of the underlying asset, 0.25 units of theunderlying asset are sold, then for small changes in the underlyingasset price, assuming all other factors are unchanged, the total changein the price of the option and the profit or loss generated by holding0.25 units of the asset are null.

Vega is the rate of change in the price of an option or other derivativein response to changes in volatility, i.e., the partial derivative ofthe option price with respect to the volatility.

Volatility Convexity is the second partial derivative of the price withrespect to the volatility, i.e. the derivative of the Vega with respectto the volatility, denoted dVega/dVol.

Straddle is a strategy, which includes buying Vanilla call and putoptions having the same strike price and the same expiration.

At-the-money Delta neutral straddle is a straddle wherein the Delta ofthe call option and the Delta of the put option have the same value withopposite sign. The buyer of the at-the-money Delta neutral straddlestrategy is automatically Delta-hedged (protected from small changes inthe price of the underlying asset).

Risk Reversal (RR) is a strategy, which includes buying a Vanilla calloption and selling a Vanilla put option with the same expiration sandthe same Delta with opposite sign. In some markets, the RR correspondsto the difference between the implied volatility of a call option and aput option with the same delta (in opposite directions). Traders in thecurrency and/or commodity option markets generally use 25delta RR, whichis the difference between the implied volatility of a 25delta calloption and a 25delta put option. Thus, 25delta RR may be calculated asfollows:25delta RR=implied Vol (25delta call)−implied Vol (25delta put)

The 25delta RR may correspond to a combination of buying a 25 delta calloption and selling a 25 delta put option. Accordingly, the 25delta RRmay be characterized by a slope of Vega of such combination with respectto spot. Thus, the price of the 25delta RR may characterize the price ofthe Vega slope, since practically the convexity of 25delta RR at thecurrent spot is close to zero. Therefore, the 25delta RR as definedabove may be used to price the slope dVega/dspot.

Strangle is a strategy of buying call and put options with the sameexpiration. In some applications, the call and put options may have thesame Delta with opposite signs. The strangle price can be presented asthe average of the implied volatility of the call and put options. Forexample:25delta strangle=0.5(implied Vol (25delta call)+implied Vol (25deltaput))

The 25delta strangle may be characterized by practically no slope ofVega with respect to spot at the current spot, but a lot of convexity,i.e., a change of Vega when the volatility changes. Therefore, it isused to price convexity.

Since the at-the-money Vol may be known, it is more common to quote thebutterfly strategy, in which one buys one unit of the strangle and sells2 units of the ATM 25 option. In some assets, the strangle/butterfly isquoted in terms of volatility. For example:25delta butterfly=0.5*(implied Vol (25delta call)++implied Vol (25deltaput))−ATM Vol

The reason it is more common to quote the butterfly rather than thestrangles is that butterfly provides a strategy with almost no Vega butsignificant convexity. Since butterfly and strangle are related throughthe ATM volatility, which may be known, they may be usedinterchangeably. The 25delta put and the 25delta call can be determinedbased on the 25delta RR and the 25delta strangle. The ATM volatility, 25delta risk reversal and/or the 25 delta butterfly may be referred to,for example, as the “Volatility Parameters”. The Volatility Parametersmay include any additional and/or alternative parameters and/or factors.

Bid/offer spread is the difference between the bid price and the offerprice of a financial derivative. In the case of options, the bid/offerspread may be expressed, for example, either in terms of volatility orin terms of the price of the option. For example, the bid/ask spread ofexchange traded options is quoted in price terms (e.g., cents, etc). Thebid/offer spread of a given option depends on the specific parameters ofthe option. In general, the more difficult it is to manage the risk ofan option, the wider is the bid/offer spread for that option.

In order to quote a price, traders typically try to calculate the priceat which they would like to buy an option (i.e., the bid side) and theprice at which they would like to sell the option (i.e., the offerside). Many traders have no computational methods for calculating thebid and offer prices, and so traders typically rely on intuition,experiments involving changing the factors of an option to see how theyaffect the market price, and past experience, which is considered to bethe most important tool of traders.

One dilemma commonly faced by traders is how wide the bid/offer spreadshould be. Providing too wide a spread reduces the ability to compete inthe options market and is considered unprofessional, yet too narrow aspread may result in losses to the trader. In determining what prices toprovide, traders need to ensure that the bid/offer spread isappropriate. This is part of the pricing process, i.e., after the traderdecides where to place the bid and offer prices, he/she needs toconsider whether the resultant spread is appropriate. If the spread isnot appropriate, the trader needs to change either or both of the bidand offer prices in order to show the appropriate spread.

Option prices that are quoted in exchanges typically have a relativelywide spread compared to their bid/ask spread in the OTC market, wheretraders of banks typically trade with each other through brokers. Inaddition the exchange price typically corresponds to small notionalamounts of options (lots). A trader may sometimes change the exchangeprice of an option by suggesting a bid price or an offer price with arelatively small amount of options. This may result in the exchangeprices being distorted in a biased way.

In contrast to the exchanges, the OTC option market has a greater“depth” in terms of liquidity. Furthermore, the options traded in theOTC market are not restricted to the specific strikes and expirationdates of the options traded in the exchanges. In addition, there aremany market makers, which quote bid/offer prices, which are totallydifferent from the bid/offer prices in the exchange.

One of the reasons that exchange prices of options are quoted with awide spread is that the prices of options corresponding to manydifferent strikes, and many different dates may change very frequently,e.g., in response to each change in the price of the underlying assets.As a result, the people that provide the bid and ask prices to theexchange have to constantly update a large number of bid and ask pricessimultaneously, e.g., each time the price of the underlying assetschanges. In order to avoid this tedious activity, it is mostly preferredto use “safe” bid and ask prices, which will not need to be frequentlyupdated.

SUMMARY

Some demonstrative embodiments include devices, systems and/or methodsof pricing financial instruments.

Although some embodiments are described herein with reference to pricinga Vanilla option, other embodiments may be implemented for pricing anyother suitable exotic option, e.g., based on the pricing of acorresponding vanilla option.

Some demonstrative embodiments include methods, devices and/or systemsimplementing a pricing model for pricing, e.g., in real time, options insubstantially all asset classes, for example, in a way that trulyreplicates the traded prices of the options, e.g., as traded in theinterbank market.

In some demonstrative embodiments, a pricing module may receive firstinput data corresponding to at least one parameter defining an option onan underlying asset and second input data corresponding to at least onecurrent market condition relating to the underlying asset.

In some demonstrative embodiments, the first input data may include anindication of at least one of a type of the option, an expiration dateof the option, a trigger for the option, and a strike of the option.

In some demonstrative embodiments, the second input data may include anindication of at least one of a spot value, a forward rate, an interestrate, a volatility, an at-the-money volatility, a delta risk reversal, adelta butterfly, a delta strangle, a 10 delta risk reversal, a 10 deltabutterfly, a 10 delta strangle, a 25 delta risk reversal, a 25 deltabutterfly, a 25 delta strangle, a caplet, a floorlet, a swap rate, asecurity lending rate, and an exchange price.

In some demonstrative embodiments, the pricing module may price theoption based on the first and second input data.

The Black Scholes (BS) model assumes that there is a single volatilityfor any maturity regardless of the strike and, that this singlevolatility, which reflects the rate of fluctuation of the price of theunderlying asset, is constant throughout the life of the option.Therefore the BS model assumes that a trader only has to constantlyre-hedge the price of the underlying asset, e.g., by always keeping theDelta amount of the underlying asset, in order to eliminate the pricerisk of the option. In reality, this assumption is not true. Typically,the volatility changes when the price of the underlying asset changes.Therefore, there is a different “volatility value” for differentstrikes. The BS model ignores the cost of rehedging the volatilitychanges.

In some demonstrative embodiments, the pricing module may consider there-hedging of two “axes”, e.g., which may be almost orthogonal to oneanother. A first “axis” may result from the fact that there is thevolatility “smile”, wherein the volatility may be affected by changes inthe price of the underlying asset price. The first axis may bere-hedged, for example, using a risk reversal strategy. The second axismay result from a Vega hedged book becoming un-hedged, e.g., when thevolatility changes. The second axis may be re-hedged, for example, usingthe strangle strategy.

In some demonstrative embodiments, the pricing module may price theoption according to a volatility smile, which may satisfy one or morepredefined criterions.

The term “volatility smile” as used herein relates to the behavior ofthe implied volatility with respect to the strike, i.e., the impliedvolatility as a function of the strike, where the implied volatility forthe ATM strike is the given ATM volatility in the market. The plot ofthe implied volatility as a function of the strike may typically show aminimum that looks like a smile, e.g., usually in equity options theminimum volatility is below the ATM strike. However, the plot of theimplied volatility as a function of the strike may have any othersuitable behavior and/or shape, e.g., different from a “U” or “smile”shape.

In some demonstrative embodiments, the volatility smile may satisfy thecriterions, for example, with respect to a pair of options forming aDelta neutral strategy, e.g., a first option including the option to bepriced and a second option representing a position opposite to aposition of a the first option and having substantially the sameabsolute delta value as the first option.

In some demonstrative embodiments, the volatility smile may satisfy thecriterions, for example, with respect to each pair of options forming aDelta neutral strategy, e.g., including a first option and a secondoption representing a position opposite to a position of a the firstoption and having substantially the same absolute delta value as thefirst option.

In some demonstrative embodiments, the volatility smile may satisfy afirst criterion relating to a sum of a first correction corresponding tothe first option and a second correction corresponding to the secondoption.

In some demonstrative embodiments, the first correction relates to adifference between a theoretical price of the first option and the priceof the first option according to the volatility smile, and the secondcorrection relates to a difference between a theoretical price of thesecond option and the price of the second option according to thevolatility smile. The theoretical value may be determined according toany suitable model, e.g., the BS model or any other model.

In some demonstrative embodiments, the first criterion may require thatthe sum of the first and second corrections is proportional to a sum offirst and second volatility convexities corresponding to the first andsecond options, respectively.

In some demonstrative embodiments, the sum of the first and secondvolatility convexities is a predefined function of a volatility of thefirst option according to the volatility smile and a volatility of thesecond option according to the volatility smile.

In some demonstrative embodiments, the second criterion may require thatthe difference between the first and second corrections is proportionalto a difference between first and second delta convexities correspondingto the first and second options, respectively.

In some demonstrative embodiments, the difference between the first andsecond delta convexities is a second predefined function of thevolatility of the first option according to the volatility smile and thevolatility of the second option according to the volatility smile.

In some demonstrative embodiments, the first criterion requires that thesum of the first and second corrections is proportional to the sum offirst and second volatility convexities according to a firstproportionality function, which is based on the delta.

In some demonstrative embodiments, the second criterion requires thatthe difference between the first and second corrections is proportionalto the difference of the first and second delta convexities according toa second proportionality function, which is based on the delta.

In some demonstrative embodiments, at least one of the first and secondproportionality functions includes a predefined combination of the deltaand one or more market-based parameters.

In some demonstrative embodiments, the pricing module may determine themarket-based parameters based on the second input data.

In some demonstrative embodiments, the first and second proportionalityfunctions are decreasing functions of delta.

In some demonstrative embodiments, the first and second criterionrequire satisfying the following equations, respectively:

${\zeta_{C}^{\Delta} + \zeta_{P}^{\Delta}} = {{{A(\Delta)} \cdot {Vega}^{\Delta}}{d_{1}^{2}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}}$${\zeta_{C}^{\Delta} - \zeta_{P}^{\Delta}} = {{{B(\Delta)} \cdot {Vega}^{\Delta}}\frac{d_{1}}{S\sqrt{t}}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}$wherein ζ_(C) ^(Δ) and ζ_(P) ^(Δ) denote the first and secondcorrections, wherein Δ denotes the delta, wherein A(Δ) and B(Δ) denotefirst and second functions of Δ, respectively, wherein Vega^(Δ) denotesa Vega of the first and second options, wherein t denotes a time toexpiration of the first option, wherein d₁ denotes a predefined functionof the time to expiration of the first option, wherein S denotes a priceof the underlying asset, and wherein σ_(K) _(Call) and σ_(K) _(Put)denote a volatility of the first option according to the volatilitysmile and a volatility of the second option according to the volatilitysmile, respectively.

In some demonstrative embodiments, the pricing module may determine avolatility of the first option based on the first and second criterions.For example, the pricing module may determine the volatility of thefirst option according to the volatility smile.

In some demonstrative embodiments, the pricing module may determine thefirst correction corresponding to the first option based on thevolatility of the first option.

In some demonstrative embodiments, the pricing module may determine theprice of the first option based on the first correction and thetheoretical price of the first option, e.g., based on a sum of thetheoretical price and the first correction.

In some demonstrative embodiments, the pricing module may determine aprice of an exotic option on the underlying asset based on thevolatility smile. For example, the pricing module may determine a priceof a vanilla option on the underlying asset, e.g., according to thevolatility smile described above, and determine the price of the exoticoption based on the price of the vanilla option.

In some demonstrative embodiments, the pricing module may provide anoutput based on the price of the first option, e.g., in real time.

In some demonstrative embodiments, the pricing module may communicatethe output via a communication network.

In some embodiments, a system may implement the pricing module toprovide price information for substantially any suitable option onsubstantially any suitable asset based on input market data. The marketdata may be easily obtained, e.g., on a real time basis. Thus, areal-time price of any desired option may be determined, e.g., based onreal time prices received from the exchanges and/or OTC market.

In some demonstrative embodiments, the price may be updated, e.g.,substantially immediately and/or automatically, for example, in responseto a change in spot prices and/or option prices. This may enable a userto automatically update prices for trading with the exchanges.

The trader may want, for example, to submit a plurality of bid and/oroffer (hereinafter “bid/offer”) prices for a plurality of options, e.g.,ten bid/offer prices for ten options, respectively. When entering thebids/offers to a quoting system, the trader may check the price, e.g.,in relation to the current spot prices, and may then submit thebids/offers to the exchange. Some time later, e.g., a second later, thespot price of the stock, which is the underlying asset of one or more ofthe options, may change. A change in the spot prices may be accompanied,for example, by changes in the volatility parameters, or may includejust a small spot change while the volatility parameters have notchanged. In response to the change in the spot price, the trader maywant to update one or more of the submitted bid/offer prices. The desireto update the bid/offer prices may occur, e.g., frequently, during tradetime.

The system implementing the pricing module according to somedemonstrative embodiments, may automatically update the bid/offer pricesentered by the trader, e.g., based on any desired criteria. For example,the system may evaluate the trader's bids/offers versus bid and offerprices of the options, which may be estimated by the system, e.g., whenthe trader submits the bid/offer prices. The system may thenautomatically recalculate the bid and/or offer prices, e.g., wheneverthe spot changes, and may automatically update the trader's bid/offerprices. The system may, for example, update one or more of the trader'sbid/offer prices such that a price difference between the bid/offerprice calculated by the pricing module and the trader's bid/offer priceis kept substantially constant. According to another example, the systemmay update one or more of the trader's bid/offer prices based on adifference between the trader's bid/offer prices and an average of bidand offer prices calculated by the pricing module. The system may updateone or more of the trader's bid/offer prices based on any other desiredcriteria.

It is noted, that a change of the spot price, e.g., of a few pips, mayresult in a change in one or more of the volatility parameters ofoptions corresponding to the spot price. It will be appreciated that thesystem according to some embodiments, may enable automatically updatingone or more option prices submitted by a trader, e.g., while taking intoaccount the change in the spot price, in one or more of the volatilityparameters, and/or in any other desired parameters, as described above.

In some demonstrative embodiments, the system may enable the trader tosubmit one or more quotes in the exchange in a form of relative pricesvs. prices determined by the pricing module. For example, the trader maysubmit quotes for one or more desired strikes and/or expiry dates. Thequotes submitted by the trader may be in any desired form, e.g.,relating to one or more corresponding prices determined by the pricingmodule. For example, the quotes submitted by the trader may be in theterms of the bid/offer prices determined by the pricing module plus twobasis points; in the terms of the mid market price determined by thepricing module minus four basis points, and the like. The system maydetermine the desired prices, for example, in real time, e.g., whenevera price change in the exchange is recorded. Alternatively, the systemmay determine the desired prices, according to any other desired timingscheme, for example, every predefined time interval, e.g., every half asecond.

A change in a spot price of a stock may result in changes in the pricesof a large number of options related to the stock. For example therecould be over 200 active options relating to a single stock and havingdifferent strikes and expiration dates. Accordingly, a massive bandwidthmay be required by traders for updating the exchange prices of theoptions in accordance with the spot price changes, e.g., in real time.This may lead the traders to submit to the exchange prices which may be“non-competitive”, e.g., prices including a “safety-margin”, since thetraders may not be able to update the submitted prices according to therate at which the spot prices, the volatility, the dividend, and/or thecarry rate may change.

Some demonstrative embodiments, may allow automatically updating of oneor more bid and/or offer prices submitted by a trader, e.g., asdescribed above. This may encourage the traders to submit with theexchange more aggressive bid and/or offer prices, since the traders mayno longer need to add the “safety margin” their prices for protectingthe traders against the frequent changes in the spot prices.Accordingly, the trading in the exchange may be more effective,resulting in a larger number of transactions. For example, a trader mayprovide the system with one or more desired volatility parameter and/orrates. The trader may request the system to automatically submit and/orupdate bid and/or offer prices on desired amounts of options, e.g.,whenever there is a significant change in the spot price and/or in thevolatility of the market. The trader may also update some or all of thevolatility parameters. The system may be linked, for example, to anautomatic decision making system, which may be able to decide when tobuy and/or sell options using the pricing module.

BRIEF DESCRIPTION OF THE DRAWINGS

For simplicity and clarity of illustration, elements shown in thefigures have not necessarily been drawn to scale. For example, thedimensions of some of the elements may be exaggerated relative to otherelements for clarity of presentation. Furthermore, reference numeralsmay be repeated among the figures to indicate corresponding or analogouselements. The figures are listed below.

FIG. 1 is a schematic illustration of a system, in accordance with somedemonstrative embodiments.

FIG. 2 is a schematic flow-chart illustration of a method, in accordancewith some demonstrative embodiments.

FIGS. 3A-3D are schematic illustrations of graphs depicting volatilitysmiles, in accordance with some demonstrative embodiments.

FIG. 4 is schematic illustration of an article of manufacture, inaccordance with some demonstrative embodiments.

DETAILED DESCRIPTION

In the following detailed description, numerous specific details are setforth in order to provide a thorough understanding of some embodiments.However, it will be understood by persons of ordinary skill in the artthat some embodiments may be practiced without these specific details.In other instances, well-known methods, procedures, components, unitsand/or circuits have not been described in detail so as not to obscurethe discussion.

Some portions of the following detailed description are presented interms of algorithms and symbolic representations of operations on databits or binary digital signals within a computer memory. Thesealgorithmic descriptions and representations may be the techniques usedby those skilled in the data processing arts to convey the substance oftheir work to others skilled in the art.

An algorithm is here, and generally, considered to be a self-consistentsequence of acts or operations leading to a desired result. Theseinclude physical manipulations of physical quantities. Usually, thoughnot necessarily, these quantities take the form of electrical ormagnetic signals capable of being stored, transferred, combined,compared, and otherwise manipulated. It has proven convenient at times,principally for reasons of common usage, to refer to these signals asbits, values, elements, symbols, characters, terms, numbers or the like.It should be understood, however, that all of these and similar termsare to be associated with the appropriate physical quantities and aremerely convenient labels applied to these quantities.

Discussions herein utilizing terms such as, for example, “processing”,“computing”, “calculating”, “determining”, “establishing”, “analyzing”,“checking”, or the like, may refer to operation(s) and/or process(es) ofa computer, a computing platform, a computing system, or otherelectronic computing device, that manipulate and/or transform datarepresented as physical (e.g., electronic) quantities within thecomputer's registers and/or memories into other data similarlyrepresented as physical quantities within the computer's registersand/or memories or other information storage medium that may storeinstructions to perform operations and/or processes.

The terms “plurality” and “a plurality” as used herein includes, forexample, “multiple” or “two or more”. For example, “a plurality ofitems” includes two or more items.

Some embodiments may include one or more wired or wireless links, mayutilize one or more components of wireless communication, may utilizeone or more methods or protocols of wireless communication, or the like.Some embodiments may utilize wired communication and/or wirelesscommunication.

Some embodiments may be used in conjunction with various devices andsystems, for example, a Personal Computer (PC), a desktop computer, amobile computer, a laptop computer, a notebook computer, a tabletcomputer, a server computer, a handheld computer, a handheld device, aPersonal Digital Assistant (PDA) device, a handheld PDA device, anon-board device, an off-board device, a hybrid device, a vehiculardevice, a non-vehicular device, a mobile or portable device, anon-mobile or non-portable device, a wireless communication station, awireless communication device, a cellular telephone, a wirelesstelephone, a Personal Communication Systems (PCS) device, a PDA devicewhich incorporates a wireless communication device, a device having oneor more internal antennas and/or external antennas, a wired or wirelesshandheld device (e.g., BlackBerry, Palm Treo), a Wireless ApplicationProtocol (WAP) device, or the like.

Some demonstrative embodiments of the present invention are describedherein in the context of a model for calculating a value, e.g., themarket value, of a financial instrument, e.g., a stock option. It shouldbe appreciated, however, that models in accordance with the inventionmay be applied to other options, financial instruments and/or markets,and the embodiments are not limited to stock options. One skilled in theart may apply the embodiments to other options and/or option-likefinancial instruments, e.g., options on interest rate futures, optionson commodities, and/or options on non-asset instruments, such as optionson the weather and/or the temperature, and the like, with variation asmay be necessary to adapt for factors unique to a given financialinstrument.

The term “financial instrument” may refer to any suitable “asset class”,e.g., Foreign Exchange (FX), Interest Rate, Equity, Commodities, Credit,weather, energy, real estate, mortgages, and the like; and/or mayinvolve more than one asset class, e.g., cross-asset, multi asset, andthe like. The term “financial instrument” may also refer to any suitablecombination of one or more financial instruments.

The term “derivative financial instrument” or “option” may refer to anysuitable derivative instruments, e.g., forwards, swaps, futures,exchange options and OTC options, which derive their value from thevalue and characteristics of one or more underlying assets.

Reference is now made to FIG. 1, which schematically illustrates a blockdiagram of a system 100, in accordance with some demonstrativeembodiments.

In some demonstrative embodiments, system 100 may include a pricingmodule (“pricing application”) 160 to price one or more derivativefinancial instruments, e.g., as described below.

In some demonstrative embodiments, system 100 includes one or more userstations or devices 102, for example, a PC, a laptop computer, a PDAdevice, and/or a terminal, to allow one or more users to price the oneor more financial assets using pricing module 160, e.g., as describedherein.

In some demonstrative embodiments, devices 102 may be implemented usingsuitable hardware components and/or software components, for example,processors, controllers, memory units, storage units, input units,output units, communication units, operating systems, applications, orthe like.

The user of device 102 may include, for example, a trader, a businessanalyst, a corporate structuring manager, a salesperson, a risk manager,a front office manager, a back office, a middle office, a systemadministrator, and the like.

In some demonstrative embodiments, system 100 may also include aninterface 110 to interface between users 102 and one or more elements ofsystem 100, e.g., pricing module 160. Interface 110 may optionallyinterface between users 102 and one or more Financial-Instrument (FI)systems and/or services 140. Services 140 may include, for example, oneor more market data services 149, one or more trading systems 147, oneor more exchange connectivity systems 148, one or more analysis services146 and/or one or more other suitable FI-related services, systemsand/or platforms.

In some demonstrative embodiments, pricing module 160 may be capable ofcommunicating, directly or indirectly, e.g., via interface 110 and/orany other interface, with one or more suitable modules of system 100,for example, one or more of FI systems 140, a database, a storage, anarchive, an HTTP service, an FTP service, an application, and/or anysuitable module capable of providing, e.g., automatically, input topricing module 160 and/or receiving output generated by pricing module160, e.g., as described herein.

In some demonstrative embodiments, pricing module 160 may be implementedas part of FI systems/services 140, as part of device 102 and/or as partof any other suitable system or module, e.g., as part of any suitableserver, or as a dedicated server.

In some demonstrative embodiments, pricing module 160 may include alocal or remote application executed by any suitable computing system183. For example, computing system 183 may include a suitable memory 187having stored-thereon pricing-application instructions 189; and asuitable processor 185 to execute instructions 189 resulting in pricingmodule 160.

In some demonstrative embodiments, computing system 183 may include ormay be part of a server to provide the functionality of pricing module160 to users 102. In other embodiments, computing system 183 may beimplemented as part of user station 102. For example, instructions 189may be downloaded and/or received by users 102 from another computingsystem, such that pricing module 160 may be locally-executed by users102. For example, instructions 189 may be received and stored, e.g.,temporarily, in a memory or any suitable short-term memory or buffer ofuser device 102, e.g., prior to being executed by a processor of userdevice 102. In other embodiments, computing system 183 may include anyother suitable computing arrangement, server and/or scheme.

In some demonstrative embodiments, computing system 183 may also executeone or more of FI systems/services 140. In other embodiments, pricingapplication 160 may be implemented separately from one or more of FIsystems/services 140.

In some demonstrative embodiments, interface 110 may be implemented aspart of pricing module 160, FI systems/services 140 and/or as part ofany other suitable system or module, e.g., as part of any suitableserver.

In some demonstrative embodiments, interface 110 may be associated withand/or included as part of devices 102. In one example, interface 110may be implemented, for example, as middleware, as part of any suitableapplication, and/or as part of a server. Interface 110 may beimplemented using any suitable hardware components and/or softwarecomponents, for example, processors, controllers, memory units, storageunits, input units, output units, communication units, operatingsystems, applications. In some demonstrative embodiments, interface 110may include, or may be part of a Web-based pricing applicationinterface, a web-site, a web-page, a stand-alone application, a plug-in,an ActiveX control, a rich content component (e.g., a Flash or Shockwavecomponent), or the like.

In some demonstrative embodiments, interface 110 may also interfacebetween users 102 and one or more of FI systems and/or services 140.

In some demonstrative embodiments, interface 110 may be configured toallow users 102 to enter commands; to define a derivative financialinstrument to be priced by pricing module 160; to define and/orstructure a trade corresponding to the derivative financial instrument;to receive a pricing of the derivative financial instrument from pricingmodule 160; to analyze the trade; to transact the trade; and/or toperform any other suitable operation.

In some demonstrative embodiments, pricing module 160 may be capable ofpricing, e.g., accurately and/or in real-time, an option, e.g., anysuitable Vanilla option, on any suitable underlying asset, e.g. optionson currencies, interest rates, commodities, equity, energy, credit,weather, and the like.

In some demonstrative embodiments, given the price of European Vanillaoptions, one can obtain the probability function, denoted P(S_(T)),which represents the probability that the price of underlying asset attime T to be S_(T), e.g., regardless of the pricing model. For example,since:

$\begin{matrix}{{{Price}_{Call} = {{df}_{R}{\int_{K}^{\infty}\ {{\mathbb{d}{S_{T}\left( {S_{T} - K} \right)}} \cdot {P\left( S_{T} \right)}}}}}{{then}\text{:}}} & (1) \\{{P\left( S_{T} \right)} = \frac{\partial^{2}{Price}_{Call}}{\partial K^{2}}} & (2)\end{matrix}$wherein Price_(call) denotes the price of a call option, df_(R) denotesa factor for time T calculated using a term currency annual interestrate R, and K denotes the strike price.

Accordingly, in some demonstrative embodiments, pricing module 160 mayuse the probability function obtained from the vanilla model tocalculate the price of any other suitable, e.g., exotic, option via, forexample, a suitable Monte Carlo simulation.

Hence, although some embodiments are described herein with reference topricing a Vanilla option, it will be appreciated that other embodimentsmay be implemented for pricing any other suitable exotic option, e.g.,based on the pricing of a corresponding vanilla option.

In some demonstrative embodiments, pricing module 160 may implement thepricing model described below for pricing, in real time, options insubstantially all asset classes in a way that truly replicates thetraded prices of the options, e.g., as traded in the interbank market.

In some demonstrative embodiments, pricing module 160 may calculate oneor more values of the volatility parameter, denoted σ=σ(K), for one ormore strikes K, e.g., for each strike K; and determine the price of theoption based on the calculated volatility parameters, for example, byapplying the Black-Scholes (BS) model, or any other suitable model, tothe determined volatility parameters, e.g., as described in detailbelow.

In some demonstrative embodiments, pricing module 160 may determine acorrection to be applied to a theoretical value of the option. Thetheoretical value may be determined according to any suitable model,e.g., the BS model or any other model.

In some demonstrative embodiments, pricing module 160 may price theoption according to a volatility smile, which may satisfy one or morepredefined criterions.

In some demonstrative embodiments, the volatility smile may satisfy thecriterions, for example, with respect to a pair of options forming aDelta neutral strategy. For example, the pair of options may include,for example, a first option, e.g., the option to be priced, and a secondoption representing a position opposite to a position of a the firstoption and substantially the same absolute delta value as the firstoption. The term “absolute delta value” as used herein relates to anabsolute of the delta. For example, first and second delta values may bethe same of they have substantially the same absolute value, regardlessof the sign.

In some demonstrative embodiments, the volatility smile may satisfy thecriterions, for example, with respect to each pair of options includinga first option and a second option representing a position opposite to aposition of a the first option and having substantially the sameabsolute delta value as the first option.

In some demonstrative embodiments, the volatility smile may satisfy afirst criterion relating to a sum of a first correction corresponding tothe first option and a second correction corresponding to the secondoption.

In some demonstrative embodiments, the first correction relates to adifference between a theoretical price of the first option and the priceof the first option according to the volatility smile, and the secondcorrection relates to a difference between a theoretical price of thesecond option and the price of the second option according to thevolatility smile, e.g., as described in detail below.

In some demonstrative embodiments, the notation d₁ may be defined asfollows:

$\begin{matrix}{d_{1} = {\frac{\log\left( {F/K} \right)}{\sigma\sqrt{t}} + {\frac{1}{2}\sigma\sqrt{t}}}} & (3)\end{matrix}$wherein F denotes the forward rate, and t denotes the time to expirationof the option.

The BS model for Vanilla call and put options may be represented usingthe notation d₁, e.g., as follows:BS ^(Call) =df _(R)(FN(d ₁)−KN(d ₁ −σ√{square root over (t)}))  (4)BS ^(Put) =df _(R)(−FN(−d ₁)+KN(−d ₁ +σ√{square root over (t)}))  (5)wherein BS^(Call) denotes the BS value of the call option, BS^(Put)denotes the BS value of the put option, and wherein N(x) denotes thecumulative normal distribution function of x, e.g., as follows:

$\begin{matrix}{{N(x)} = {\int_{- \infty}^{x}{\frac{{\mathbb{e}}^{{- t^{2}}/2}}{\sqrt{2\;\pi}}\ {\mathbb{d}t}}}} & (6)\end{matrix}$

The BS values BS^(Call) and BS^(Put) according to Equations 4 and 5 mayrepresent the respective prices of a call option to buy and a put optionto sell one unit of asset at the predetermined strike price K at apredetermined expiration date t.

The delta of a call option and a put option, denoted Δ_(Call) andΔ_(Put), respectively, i.e., the rate of change in the price of the calloption and the put option, respectively, in response to changes in theprice of the underlying asset, may be determined as follows:Δ_(Call) =df _(L) ·N(d ₁)  (7)Δ_(Put) =−df _(L) ·N(−d ₁)  (8)wherein df_(L) is a discount factor, which is calculated using a baseannual interest like rate L. For example, in stocks L is the dividendrate, in commodities L is the carry or convenience rate, and incurrencies L is the base currency interest rate. The discount factorsdf_(L) and df_(R) may be related by the formula F=S·df_(L)/df_(R),wherein S is the current price (rate) of the asset.

Accordingly, a call option and a put option having the same deltasatisfy the following condition:d ₁(K _(Call))=−d ₁(K _(Put))  (9)wherein K_(Call) denotes the strike of the call option, and K_(Put)denotes the strike of the put option.

In some demonstrative embodiments, the rate of change, denoted Vega, inthe price of an option in response to changes in the volatility may bedefined as follows:Vega=df _(L) ·S√{square root over (t)}·n(d ₁)  (10)wherein n(t) denotes the normal probability density function of t, e.g.,as follows:

$\begin{matrix}{{n(t)} = \frac{{\mathbb{e}}^{{- t^{2}}/2}}{\sqrt{2\;\pi}}} & (11)\end{matrix}$

In some demonstrative embodiments, a first strategy (“same delta RiskReversal”) may be defined to include buying a call option and selling aput option having a delta of the same value and an opposite sign of thedelta of the call option; and a second strategy (“same delta strangle”)may be defined to include buying a call option and buying a put optionhaving a delta of the same value and an opposite sign of the delta ofthe call option. According to Equations 9 and 10, a put option and acall option having the same delta with opposite signs may also have thesame Vega (hereinafter referred to as “having the same delta”).Accordingly, the derivatives of the Vega of the first and secondstrategies may satisfy the following conditions:

$\begin{matrix}{\begin{matrix}{{\frac{\partial{Vega}_{Call}^{\Delta}}{\partial S} - \frac{\partial{Vega}_{Put}^{\Delta}}{\partial S}} = {{{- {df}_{L}} \cdot {n\left( d_{1} \right)} \cdot {d_{1}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}} =}} \\{= {{- {Vega}^{\Delta}}\frac{d_{1}}{S\sqrt{t}}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}}\end{matrix}{{and}\text{:}}} & (12) \\\begin{matrix}{{\frac{\partial{Vega}_{Call}^{\Delta}}{\partial\sigma} + \frac{\partial{Vega}_{Put}^{\Delta}}{\partial\sigma}} = {{{df}_{L} \cdot S}{\sqrt{t} \cdot {N\left( d_{1} \right)} \cdot}}} \\{{d_{1}^{2}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)} =} \\{= {{Vega}^{\Delta}{d_{1}^{2}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}}}\end{matrix} & (13)\end{matrix}$wherein Vega_(Call) ^(Δ) and Vega_(Put) ^(Δ) denote the value of Vegafor the call and put options, respectively, on the same underlying assetand having the same Delta.

The BS model assumes that there is a single volatility for any maturityregardless of the strike and, that this single volatility, whichreflects the rate of fluctuation of the price of the underlying asset,is constant throughout the life of the option. Therefore the BS modelassumes that a trader only has to constantly re-hedge the price of theunderlying asset (by always keeping the Delta amount of the underlyingasset) in order to eliminate the price risk of the option. It is wellknown that in reality this assumption is not true. Typically thevolatility changes when the price of the underlying asset changes.Therefore, there is a different “volatility value” for differentstrikes. The BS model ignores the cost of rehedging the volatilitychanges.

In some demonstrative embodiments, pricing module 160 may implement apricing model (“the Gershon model”), which may at least partially fixthis flaw of the BS model, e.g., as described herein.

In some demonstrative embodiments, the Gershon model may consider there-hedging of two “axes”, e.g., which may be almost orthogonal to oneanother. A first “axis” may result from the fact that there is thevolatility “smile”, wherein the volatility may be affected by changes inthe price of the underlying asset price. The first axis may be re-hedgedusing the risk reversal. The second axis may result from a Vega hedgedbook becoming un-hedged, e.g., when the volatility changes. The secondaxis may be re-hedged using the strangle.

In some demonstrative embodiments, the Delta neutral straddle strategymay be defined to include call and put options with the same strike,denoted K₀, at which:Δ_(Call) ^((K) ⁰ ⁾=−Δ_(Put) ^((K) ⁰ ⁾  (14)

Therefore, d₁=0 and

$K_{0} = {F\;{{\mathbb{e}}^{\frac{1}{2}\sigma^{2}t}.}}$According to this definition, the Delta, denoted Δ₀, of the call or theput of the Delta neutral straddle strategy is:Δ₀ =df _(L)/2  (15)

The volatility, denoted σ₀, may be defined as the volatility, which, ifsubstituted in the BS model for the strike K_(o), yields the marketprice of the option with the strike K_(o).

In some demonstrative embodiments, the Gershon model may implement acorrection (“Zeta”), denoted ζ, to be added to the value of an optiondetermined according to the BS model (“the BS value”), e.g., adifference between the value of the option according to the Gershonmodel (“the market price”) at the strike K, and the BS value with thevolatility σ₀ that is used in the BS model for the strike of theat-the-money Delta neutral straddle. The correction ζ, may be defined,for example, as follows:ζ=Market price(K)−BS(K)  (16)

In some demonstrative embodiments, the Gershon model may assume thatσ_(o) is the BS volatility such that BS^(C) (σ₀,K₀) generates thecorrect market price for the strike K₀.

In some demonstrative embodiments, the correction, denoted ζ_(C), viathe function σ(K) to the call option may be represented as follows:ζ_(C)(K)=BS ^(Call)(σ_(K) ,K)−BS ^(Call)(σ₀ ,K)  (17)

In some demonstrative embodiments, the correction, denoted ζ_(p), viathe function σ(K) to the put option may be represented as follows:ζ_(P)(K)=BS ^(Put)(σ_(K) ,K)−BS ^(Put)(σ₀ ,K)  (18)

By definition of the corrections ζ_(C) and ζ_(P), the corrections ζ_(C)and ζ_(p) at K₀ satisfy ζ_(C) ^(Δ) ⁰ =ζ_(P) ^(Δ) ⁰ =0. It is noted, thatsince buying a call option together with selling a put option with thesame strike is equivalent to entering a forward deal at a forward rateequal the strike, the value of the correction ζ_(C) for the call optionis identical to the value of the correction ζ_(P) for the put optionwith the same strike, e.g., regardless of the pricing model.

In some demonstrative embodiments, the sum of the first and secondcorrections may be proportional to the sum of first and secondvolatility convexities corresponding to the first and second options,respectively, according to a first proportionality function, which isbased on the delta.

For example, the correction of the strangle strategy, which is the sumof the corrections ζ corresponding to the call and put options on thesame underlying asset and having the same Delta, may be proportional tothe sum of the derivatives of Vega with respect to the volatility, inaccordance with Equation 13, e.g., as follows:

$\begin{matrix}{{\zeta_{C}^{\Delta} + \zeta_{P}^{\Delta}} = {{{A(\Delta)} \cdot {Vega}^{\Delta}}{d_{1}^{2}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}}} & (19)\end{matrix}$wherein A(Δ) denotes a first proportionality function of Δ, e.g., asdescribed below.

In some demonstrative embodiments, the difference between the first andsecond corrections may be proportional to the difference of first andsecond delta convexities corresponding to the first and second options,respectively, according to a second proportionality function, which isbased on the delta.

The term “Delta convexity” as used herein may relate to a derivative ofVega with respect to the spot S.

For example, the correction of the risk-reversal strategy, which is thedifference between the corrections corresponding to the call and putoptions on the same underlying asset and having the same Delta, may beproportional to the difference of the derivatives of Vega with respectto S, for example, in accordance with Equation 12, e.g., as follows:

$\begin{matrix}{{\zeta_{C}^{\Delta} - \zeta_{P}^{\Delta}} = {{{B(\Delta)} \cdot {Vega}^{\Delta}}\frac{d_{1}}{S\sqrt{t}}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}} & (20)\end{matrix}$wherein B(Δ) denotes a second proportionality function of Δ, e.g., asdescribed below.

In some demonstrative embodiments, the functions A(Δ) and B(Δ) aredecreasing functions of Δ and have to satisfy market conditions, e.g.,as described in detail below. The functions A(Δ) and B(Δ) may depend onany suitable parameters and/or factors, e.g., the time to expiration t,and the like.

In some demonstrative embodiments, the proportionality functions A(Δ)and/or B(Δ) may include a predefined combination of the delta and one ormore market-based parameters.

In some demonstrative embodiments, module 160 may determine themarket-based parameters based on the second input data.

In some demonstrative embodiments, the market-based parameters of theproportionality functions A(Δ) and/or B(Δ) may depend on the maturity fthe option and/or any other suitable factor, e.g., except for thestrike, d or the volatility. In other embodiments, the market-basedparameters may depend on any other suitable factor.

In some demonstrative embodiments, the proportionality functions A(Δ)and/or B(Δ) may be decreasing functions of delta.

In some demonstrative embodiments, the market date may relate to aplurality of option prices may be obtained from the market. Themarket-based parameters of the proportionality functions A(Δ) and/orB(Δ) may be determined by fitting the proportionality functions A(Δ)and/or B(Δ) to the market data. Equations 19 and 20 may then be usedwith the determined proportionality functions A(Δ) and/or B(Δ) to pricean option of any suitable strike, e.g., as described herein.

In one embodiment, in the market of currency options (FX), the 25Δ riskreversal and 25Δ butterfly may be traded. Hence, the functions A(Δ)and/or B(Δ) may be determined such that Equations 19 and 20 satisfy thetraded 25Δ risk reversal and 25Δ butterfly of the market. Optionally, anumber of free parameters in the functions A(Δ) and/or B(Δ) may beselected to satisfy additional conditions. For example, in some currencypairs, additional delta values may be traded in the market, e.g., the10Δ risk reversal and/or the 10Δ butterfly. Accordingly, the functionsA(Δ) and/or B(Δ) may be determined such that Equations 19 and 20 satisfythe 10Δ risk reversal and/or the 10Δ butterfly of the market.

In another embodiment, in the market of Equity, the functions A(Δ)and/or B(Δ) may be determined depending on a plurality of strike pricesof options traded in the market. For example, the functions A(Δ) and/orB(Δ) may be determined by requiring a best fit between the pricesaccording to Equations 19 and 20 and between the exchange prices of theplurality of strikes and/or depending on suitable fixed strikes that aremore liquid.

In another embodiment, in the interest rates caps and floors market, thefunctions A(Δ) and/or B(Δ) may be determined based on thecaplets/floorlets to generate the correct market prices for the caps andfloors.

In another embodiment, in the Swaptions market the functions A(Δ) and/orB(Δ) may be determined based on a best fit for swaption prices of thesame swap length and the same expiration with different strikes (fixedrate of the swap), which are typically denoted by a difference, in basispoints, from the at-the-money forward strike.

In one example, the functions A(Δ) and B(Δ) may defined as follows,e.g., for Δ₀>Δ:A(Δ)=α_(1e) ^(−β) ¹ ^((Δ) ⁰ ^(-Δ))  (21)B(Δ)=α_(2e) ^(−β) ² ^((Δ) ⁰ ^(-Δ))  (22)wherein α₁,α₂,β₁,β₂ denote four respective market parameters to bedetermined, e.g., based on the traded market data.

It is noted, that there may be no need to handle the situation of Δ₀<Δ,e.g., since Equations 19 and 20 are simultaneously solved for both calland put options, e.g., as described below, and, therefore, Δ_(o) is themaximal Delta to be handled.

In some demonstrative embodiments, pricing module 160 may implement theGershon model to determine the volatility σ_(K) _(Call) corresponding toa given a strike K_(Call)>K₀, for example, by solving the followingequation:

$\begin{matrix}{{\zeta_{P}\left( {\sigma_{K_{Put}},K_{Put}} \right)} = {{\zeta_{C}\left( {\sigma_{K_{Call}},K_{Call}} \right)} \cdot \frac{1 - {{{{B(\Delta)}/{A(\Delta)}} \cdot S}{\sqrt{t} \cdot d_{1}}}}{1 + {{{{B(\Delta)}/{A(\Delta)}} \cdot S}\sqrt{t \cdot d_{1}}}}}} & (23)\end{matrix}$wherein σ_(K) _(Put) denotes the volatility of a put option at thestrike K_(Put)<K₀, which may be determined, e.g., based on Equations 19and 20, for example, as follows:

$\begin{matrix}{\sigma_{K_{Put}} = \left( {\frac{2\;{\zeta_{C}\left( {K_{Call},\sigma_{K_{Call}}} \right)}}{\left( {{Vega}^{\Delta}{d_{1}^{2}\left( {{A(\Delta)} + \frac{B(\Delta)}{S{\sqrt{t} \cdot d_{1}}}} \right)}} \right)} - \frac{1}{\sigma_{K_{Call}}}} \right)^{- 1}} & (24)\end{matrix}$wherein the strike K_(Put) may be determined, e.g., based on Equations 3and 9, for example, as follows:

$\begin{matrix}{K_{Put} = {F\;{\mathbb{e}}^{({{d_{1}\sigma_{K_{Put}}\sqrt{t}} + {\frac{1}{2}{\sigma_{K_{Put}}^{2} \cdot t}}})}}} & (25)\end{matrix}$and wherein, as mentioned above:

$\begin{matrix}{{{\Delta = {\Delta\left( d_{1} \right)}};}{d_{1} = {\frac{\log\left( {F/K_{Call}} \right)}{\sigma_{K_{Call}}\sqrt{t}} + {\frac{1}{2}\sigma_{K_{Call}}\sqrt{t}}}}} & (26)\end{matrix}$

In some demonstrative embodiments, Equation 23 may be solved, e.g.,using any suitable numerical method or algorithm, to determine the valueof σ_(K) _(Call) .

Additionally or alternatively, pricing module 160 may implement theGershon model to determine the volatility σ_(K) _(Putl) corresponding tothe given strike K_(Put)<K₀, since Equations 19 and 20 are symmetricwith respect to the call and put options. For example, the volatilityσ_(K) _(Putl) may be determined explicitly by solving the followingequation:

$\begin{matrix}{{\zeta_{C}\left( {\sigma_{K_{Call}},K_{Call}} \right)} = {{\zeta_{P}\left( {\sigma_{K_{Put}},K_{Put}} \right)} \cdot \frac{1 - {{{{B(\Delta)}/{A(\Delta)}} \cdot S}{\sqrt{t} \cdot d_{1}}}}{1 + {{{{B(\Delta)}/{A(\Delta)}} \cdot S}\sqrt{t \cdot d_{1}}}}}} & (27)\end{matrix}$wherein, e.g., based on Equations 19 and 20:

$\begin{matrix}{\sigma_{K_{Call}} = \left( {\frac{2\;{\zeta_{P}\left( {K_{Put},\sigma_{K_{Put}}} \right)}}{\left( {{Vega}^{\Delta}{d_{1}^{2}\left( {{A(\Delta)} - \frac{B(\Delta)}{S{\sqrt{t} \cdot d_{1}}}} \right)}} \right)} - \frac{1}{\sigma_{K_{Put}}}} \right)^{- 1}} & (28)\end{matrix}$wherein, e.g., based on Equation 3:

$\begin{matrix}{K_{Call} = {Fe}^{({{{- d_{1}}\sigma_{K_{Call}}\sqrt{t}} + {\frac{1}{2}{\sigma_{K_{Call}}^{2} \cdot t}}})}} & (29)\end{matrix}$and wherein, as mentioned above:

$\begin{matrix}{{- d_{1}} = {\frac{\log\left( {F/K_{Put}} \right)}{\sigma_{K_{Put}}\sqrt{t}} + {\frac{1}{2}\sigma_{K_{Put}}\sqrt{t}}}} & (30)\end{matrix}$

In some demonstrative embodiments, Equation 27 may be solved, e.g.,using any suitable numerical method or algorithm, to determine the valueof σ_(K) _(Put) .

Following is an example, in accordance with one embodiment, ofdetermining the functions A(Δ) and B(Δ) using the 25 delta strikes.However, it will be appreciated that in other embodiments the functionsA(Δ) and B(Δ) may be determined in any other suitable manner, e.g.,using any suitable data and/or parameters.

In some markets, e.g., the currencies and/or commodities markets, the 25delta strikes may be traded. Accordingly, the values of σ_(25ΔC) andσ_(25ΔP) for the call and put options, respectively, may be receivedfrom the market. The values of the functions A(Δ) and B(Δ) at the 25delta strikes may be determined, for example, since at the 25 delta0.25=df_(L)N(d₁), then:

$\begin{matrix}{d_{1} = {N^{- 1}\left( {0.25/{df}_{L}} \right)}} & (31) \\{{Vega}^{25\;\Delta} = {{df}_{L}S{\sqrt{t} \cdot {n\left( {N^{- 1}\left( {0.25/{df}_{L}} \right)} \right)}}}} & (32) \\\begin{matrix}{{A\left( {\Delta = 25} \right)} = \frac{\left( {\zeta_{C}^{25\;\Delta} + \zeta_{P}^{25\;\Delta}} \right)}{\begin{matrix}{{df}_{L}S{\sqrt{t} \cdot {n\left( {N^{- 1}\left( {0.25/{df}_{L}} \right)} \right)} \cdot}} \\{\left( {N^{- 1}\left( {0.25/{df}_{L}} \right)} \right)^{2}\left( {\frac{1}{\sigma_{25\;\Delta\; C}} + \frac{1}{\sigma_{25\;\Delta\; P}}} \right)}\end{matrix}}} \\{= {\alpha_{1}{\mathbb{e}}^{- {\beta_{1}{({{0.5\;{df}_{L}} - 0.25})}}}}}\end{matrix} & (33) \\\begin{matrix}{{B\left( {\Delta = 25} \right)} = \frac{\left( {\zeta_{C}^{25\;\Delta} + \zeta_{P}^{25\;\Delta}} \right)}{\begin{matrix}{{df}_{L} \cdot {n\left( {N^{- 1}\left( {0.25/{df}_{L}} \right)} \right)} \cdot} \\{\left( {N^{- 1}\left( {0.25/{df}_{L}} \right)} \right)\left( {\frac{1}{\sigma_{25\;\Delta\; C}} + \frac{1}{\sigma_{25\;\Delta\; P}}} \right)}\end{matrix}}} \\{= {\alpha_{2}{\mathbb{e}}^{- {\beta_{2}{({{0.5\;{df}_{L}} - 0.25})}}}}}\end{matrix} & (34)\end{matrix}$

The values, denoted BS(25Δcall) and BS(25Δput), of the respective 25delta call and put options according to the BS model may be determined,for example, by substituting d1 of Equation 31 into Equations 4 and 5,as follows:

$\begin{matrix}{{{BS}\left( {25\;\Delta\;{call}} \right)} = {{0.25\; S} - {{K_{25\;\Delta\; C} \cdot {df}_{R}}{N\left( {{N^{- 1}\left( {0.25/{df}_{L}} \right)} - {\sigma_{25\;\Delta\; c} \cdot \sqrt{t}}} \right)}}}} & (35) \\{{{BS}\left( {25\;\Delta\;{put}} \right)} = {{{- 0.25}\; S} + {{K_{25\;\Delta\; P} \cdot {df}_{R}}{N\left( {{N^{- 1}\left( {0.25/{df}_{L}} \right)} - {\sigma_{25\;{\Delta p}} \cdot \sqrt{t}}} \right)}}}} & (36) \\{\mspace{79mu}{{wherein}\text{:}}} & \; \\{\mspace{79mu}{K_{put}^{25\;\Delta} = {F\;{\mathbb{e}}^{({{{{N^{- 1}{({0.25/{df}_{L}})}} \cdot \sigma_{25\;\Delta\; p}}\sqrt{t}} + {\frac{1}{2}\sigma_{25\;\Delta\; p}^{2}t}})}}}} & (37) \\{\mspace{79mu}{K_{call}^{25\;\Delta} = {F\;{\mathbb{e}}^{- {({{{{N^{- 1}{({0.25/{df}_{L}})}} \cdot \sigma_{25\;\Delta\; c}}\sqrt{t}} - {\frac{1}{2}\sigma_{25\;\Delta\; c}^{2}t}})}}}}} & (38) \\{\mspace{79mu}{{Accordingly}\text{:}}} & \; \\{{{BS}\left( {25\;\Delta\;{call}} \right)}=={{0.25\; S} - {{df}_{R}F\;{{\mathbb{e}}^{- {({{{{N^{- 1}{({0.25/{df}_{L}})}} \cdot \sigma_{25\;\Delta\; c}}\sqrt{t}} - {\frac{1}{2}\sigma_{25\;\Delta\; c}^{2}t}})}} \cdot {N\left( {{N^{- 1}\left( {0.25/{df}_{L}} \right)} - {\sigma_{25\;\Delta\; c}^{2} \cdot t}} \right)}}}}} & (39) \\{{{BS}\left( {25\;\Delta\;{put}} \right)}=={{{- 0.25}\; S} + {{df}_{R}F{\quad\;{{\mathbb{e}}^{- {({{{{- {N^{- 1}{({0.25/{df}_{L}})}}} \cdot \sigma_{25\;\Delta\; c}}\sqrt{t}} - {\frac{1}{2}\sigma_{25\;\Delta\; p}^{2}t}})}} \cdot {N\left( {{N^{- 1}\left( {0.25/{df}_{L}} \right)} + {\sigma_{25\;\Delta\; P}^{2} \cdot t}} \right)}}}}}} & (40)\end{matrix}$

The corrections ζ_(25ΔC) and ζ_(25ΔP) corresponding to the call and putoptions at the 25 delta strike may be determined, e.g., based on theabove-listed Equations, for example, as follows:

The values of d₁ corresponding to the call and put options at the 25delta strike may be determined, for example, as follows:

$\begin{matrix}\begin{matrix}{d_{1_{25\;\Delta\; C}}^{0} = {\frac{{\log\left( {F/K} \right)} + {\frac{1}{2}\sigma_{0}^{2}t}}{\sigma_{0}\sqrt{t}} =}} \\{= {{{N^{- 1}\left( {0.25/{df}_{L}} \right)}\frac{\sigma_{25\;\Delta\; C}}{\sigma_{0}}} + {\frac{1}{2}\left( {\sigma_{0} - \frac{\sigma_{25\;\Delta\; C}^{2}}{\sigma_{0}}} \right)\sqrt{t}}}}\end{matrix} & (41) \\{d_{1_{25\;\Delta\; P}}^{0} = {{{- {N^{- 1}\left( {0.25/{df}_{L}} \right)}}\frac{\sigma_{25\;\Delta\; P}}{\sigma_{0}}} + {\frac{1}{2}\left( {\sigma_{0} - \frac{\sigma_{25\;\Delta\; P}^{2}}{\sigma_{0}}} \right)\sqrt{t}}}} & (42)\end{matrix}$and the corrections ζ_(25ΔC) and ζ_(25Δp) may be determined bysubtracting the BS value for σ₀ from the values BS(25Δcall) andBS(25Δput), respectively, for example, as follows:

$\begin{matrix}{\zeta_{25\;\Delta\; C} - {{df}_{R}{F\left\lbrack {{0.25/{df}_{L}} - {{{N\left( {{{N^{- 1}\left( {0.25/{df}_{L}} \right)} \cdot \frac{\sigma_{25\;\Delta\; c}}{\sigma_{0}}} + {\frac{1}{2}\left( {\sigma_{0} - {\sigma_{25\;\Delta\; c}^{2}/\sigma_{0}}} \right)\sqrt{t}}} \right)}--}{{\mathbb{e}}^{- {({{{{- {N^{- 1}{({0.25/{df}_{L}})}}} \cdot \sigma_{25\;\Delta\; c}}\sqrt{t}} - {\frac{1}{2}\sigma_{25\;\Delta\; c}^{2}t}})}} \cdot \begin{pmatrix}{{N\left( {{N^{- 1}\left( {0.25/{df}_{L}} \right)} - {\sigma_{25\;\Delta\; C}^{2}t}} \right)}--} \\{N\left( {{{N^{- 1}\left( {0.25/{df}_{L}} \right)} \cdot \frac{\sigma_{25\;\Delta\; C}^{2}}{\sigma_{0}}} - {\frac{1}{2}\left( {\sigma_{0} + {\sigma_{25\;\Delta\; c}^{2}/\sigma_{0}}} \right)\sqrt{t}}} \right)}\end{pmatrix}}}} \right\rbrack}}} & (43) \\{\zeta_{25\;\Delta\; P} = {{df}_{R}{F\left\lbrack {{{- 0.25}/{df}_{L}} + {{{N\left( {{{N^{- 1}\left( {0.25/{df}_{L}} \right)} \cdot \frac{\sigma_{25\;\Delta\; P}}{\sigma_{0}}} - {\frac{1}{2}\left( {\sigma_{0} - {\sigma_{25\;\Delta\; p}^{2}/\sigma_{0}}} \right)\sqrt{t}}} \right)}--}{{\mathbb{e}}^{({{{{- {N^{- 1}{({0.25/{df}_{L}})}}} \cdot \sigma_{25\;\Delta\; P}}\sqrt{t}} + {\frac{1}{2}\sigma_{25\;\Delta\; p}^{2}t}})} \cdot \cdot \begin{pmatrix}{{N\left( {{N^{- 1}\left( {0.25/{df}_{L}} \right)} + {\sigma_{25\;\Delta\; P}^{2}t}} \right)}--} \\{N\left( {{{N^{- 1}\left( {0.25/{df}_{L}} \right)}\frac{\sigma_{25\;\Delta\; P}}{\sigma_{0}}} - {\frac{1}{2}\left( {\sigma_{0} + {\sigma_{25\;\Delta\; p}^{2}/\sigma_{0}}} \right)\sqrt{t}}} \right)}\end{pmatrix}}}} \right\rbrack}}} & (44)\end{matrix}$

The relationships of α₁(β₁) and α₂(β₂) may be determined, for example,based on Equations 19, 20, 21, 22, 43 and 44. One or more additionalparameters of the functions A(Δ) and B(Δ) may be determined based on oneor more additional parameters, e.g., the σ_(10Δcall) and/or σ_(10Δput)parameters.

Following is an example, in accordance with some demonstrativeembodiments, of a method of solving Equations 19 and 20, for example, byrepresenting elements of equations 19 and 20 in terms of the notationd₁. However, it will be appreciated that in other embodiments Equations19 and 20 may be solved in any other suitable manner, e.g., using anysuitable representation, notation and/or any other solving method and/oralgorithm.

In some demonstrative embodiments, Equations 19 and 20 may be rewrittenas follows, for example, using the definition of Vega, e.g., accordingto Equation 10:

$\begin{matrix}{{\zeta_{C}^{\Delta} + \zeta_{P}^{\Delta}} = {{A\left( d_{1} \right)}{df}_{R}F\sqrt{t}{n\left( d_{1} \right)}{d_{1}^{2}\left( {\frac{1}{\sigma_{K_{call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}}} & (45) \\{{\zeta_{C}^{\Delta} - \zeta_{P}^{\Delta}} = {{B\left( d_{1} \right)}{df}_{L}{n\left( d_{1} \right)}{d_{1}\left( {\frac{1}{\sigma_{K_{call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}}} & (46)\end{matrix}$wherein A(d₁) and B(d₁) denote first and second proportionalityfunctions of d₁. The functions A(d₁) and B(d₁) may include one or moremarket-based parameters, which may be determined based on the marketdata, e.g., as described above.

In some demonstrative embodiments, a first combination of Equations 45and 46, for example, a sum of Equations 45 and 46 may yield a firstcombined Equation, e.g., as follows:

$\begin{matrix}{\zeta_{C}^{\Delta} = {\frac{1}{2}\left( {{{A\left( d_{1} \right)}{df}_{R}F\sqrt{t}d_{1}} + {{B\left( d_{1} \right)}{df}_{L}}} \right){n\left( d_{1} \right)}{d_{1}\left( {\frac{1}{\sigma_{K_{call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}}} & (47)\end{matrix}$

In some demonstrative embodiments, the volatility σ_(K) _(Call) may berepresented as a function of the notation d₁. For example, the followingrepresentation of the volatility σ_(K) _(Call) may be achieved, forexample, by rearranging Equation 26, e.g., since for a call optionK_(C)>K₀=Fe^(σ) ⁰ ² ^(t):

$\begin{matrix}{{\sqrt{t}\sigma_{K_{Call}}} = {\sqrt{{2\;\log\frac{K_{Call}}{F}} + d_{1}^{2}} + d_{1}}} & (48) \\{\frac{1}{\sqrt{t}\sigma_{K_{Call}}} = \frac{d_{1} - \sqrt{{2\;\log\frac{K_{Call}}{F}} + d_{1}^{2}}}{2\;\log\frac{F}{K_{Call}}}} & (49)\end{matrix}$

In some demonstrative embodiments, the volatility σ_(K) _(Put) may berepresented as a function of ζ_(C) and d₁. For example, the volatilityσ_(K) _(Put) may be represented as follows, e.g., by rearrangingEquation 47 using Equations 48 and 49:

$\begin{matrix}{\frac{1}{\sigma_{K_{Put}}} = {\frac{2\;\zeta_{C}^{\Delta}}{\left( {{A\left( d_{1} \right){df}_{R}F\sqrt{t}d_{1}} + {{B\left( d_{1} \right)}{df}_{L}}} \right){n\left( d_{1} \right)}d_{1}} - \frac{\sqrt{t}\left( {d_{1} - \sqrt{{2\;\log\frac{K_{Call}}{F}} + d_{1}^{2}}} \right)}{2\;\log\frac{F}{K_{Call}}}}} & (50) \\{\frac{1}{\sigma_{K_{Put}}} = \frac{2\;\log\frac{F}{K_{Call}}\left( {{A\left( d_{1} \right){df}_{R}F\sqrt{t}d_{1}} + {{B\left( d_{1} \right)}{df}_{L}}} \right){n\left( d_{1} \right)}d_{1}}{\begin{matrix}{{4\;\log\frac{F}{K_{Call}}\zeta_{C}^{\Delta}} -} \\{\begin{pmatrix}{{A\left( d_{1} \right){df}_{R}F\sqrt{t}d_{1}} +} \\{B\left( d_{1} \right){df}_{L}}\end{pmatrix}n\left( d_{1} \right)d_{1}\sqrt{t}\left( {d_{1} - \sqrt{{2\;\log\frac{K_{Call}}{F}} + d_{1}^{2}}} \right)}\end{matrix}}} & (51)\end{matrix}$

In some demonstrative embodiments, a second combination of Equations 45and 46 may yield a second combined Equation. For example, Equation 46may be subtracted from Equation 45 and rearranged, e.g., as follows:

$\begin{matrix}{\zeta_{P}^{\Delta} = {\left( {{A\left( d_{1} \right){df}_{R}F\sqrt{t}d_{1}} - {{B\left( d_{1} \right)}{df}_{L}}} \right){n\left( d_{1} \right)}{d_{1}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}}} & (52)\end{matrix}$

In some demonstrative embodiments, the corrections ζ_(C) ^(Δ) and/orζ_(P) ^(Δ) may be represented as a function of d₁, K and σ₀. Forexample, the corrections ζ_(C) ^(Δ) and/or ζ_(P) ^(Δ) may be representedas follows, e.g., by combining and rearranging Equations 4, 5, 17, 18,25, 48 and/or 49:

$\begin{matrix}\begin{matrix}{\zeta_{C}^{\Delta} = {\zeta_{C}^{\Delta}\left( {K_{Call},d_{1},\sigma_{0}} \right)}} \\{= {{{df}_{R}{N\left( d_{1} \right)}} - {{df}_{R}K_{Call}{{N\left( {- \sqrt{{2\;\log\frac{K_{Call}}{F}} + d_{1}^{2}}} \right)}--}}}} \\{{{df}_{R}{N\left( {\frac{\log\frac{F}{K_{Call}}}{\sigma_{0}\sqrt{t}} + {\frac{1}{2}\sigma_{0}^{2}\sqrt{t}}} \right)}} +} \\{{df}_{R}K_{Call}{N\left( {\frac{\log\frac{F}{K_{Call}}}{\sigma_{0}\sqrt{t}} + {\frac{1}{2}\sigma_{0}\sqrt{t}}} \right)}}\end{matrix} & (53) \\\begin{matrix}{\zeta_{P}^{\Delta} = {\zeta_{P}^{\Delta}\left( {K_{Call},d_{1},\sigma_{0}} \right)}} \\{= {{{df}_{R}{F\left( {\mathbb{e}}^{{d_{1}\sqrt{t}\sigma_{K_{Putt}}} + {\frac{1}{2}\sigma_{K_{Putt}}^{2_{t}}}} \right)}{N\left( {d_{1} + {\sigma_{K_{Putt}}\sqrt{t}}} \right)}} - {{N\left( d_{1} \right)}--}}} \\{{df}_{R}{F\left( {{{\mathbb{e}}^{({{d_{1}\sqrt{t}\sigma_{K_{Putt}}} + {\frac{1}{2}\sigma_{K_{Putt}}^{2_{t}}}})}{N\left( {\frac{{2\; d_{1}\sqrt{t}\sigma_{K_{Putt}}} + \sigma_{K_{Putt}}^{2_{t}}}{2\;\sigma_{0}\sqrt{t}} + \frac{\sigma_{0}\sqrt{t}}{2}} \right)}} -} \right.}} \\\left. {N\left( {\frac{{2\; d_{1}\sqrt{t}\sigma_{K_{Putt}}} + \sigma_{K_{Putt}}^{2_{t}}}{2\;\sigma_{0}\sqrt{t}} + \frac{\sigma_{0}\sqrt{t}}{2}} \right)} \right)\end{matrix} & (54)\end{matrix}$wherein, for example, the volatility σ_(K) _(Put) may be replacedaccording to Equation 51.

In some demonstrative embodiments, the value of d₁ may be determined,e.g., using any suitable numeric method, for example, by requiring thatEquation 52 is equal to Equation 54, e.g., as described below.

In some demonstrative embodiments, a method of determining the value ofd₁ may include selecting an initial value for d₁.

In some demonstrative embodiments, the method of determining the valueof d₁ may include determining the value of the correction ζ_(C) ^(Δ)using the value of d₁, e.g., according to Equation 53.

In some demonstrative embodiments, the method of determining the valueof d₁ may include determining the value of the volatility σ_(K) _(Put)may using the value of d₁ and the determined correction ζ_(C) ^(Δ),e.g., according to Equation 51.

In some demonstrative embodiments, the method of determining the valueof d₁ may include determining the value of the correction ζ_(P) ^(Δ)using the value of d₁ and the determined volatility σ_(K) _(Put) , e.g.,according to Equation 54.

In some demonstrative embodiments, the method of determining the valueof d₁ may include substituting the determined value of the correctionζ_(P) ^(Δ) into Equation 52 and determining whether or not thedetermined value of the correction ζ_(P) ^(Δ) satisfies Equation 52.

In some demonstrative embodiments, if, for example, the determined valueof the correction ζ_(P) ^(Δ) does not satisfy Equation 52, then anothervalue of d₁ may be selected and the determining of the value of thecorrection ζ_(C) ^(Δ), determining the value of the volatility σ_(K)_(Put) , determining the value of the correction ζ_(P) ^(Δ) anddetermining whether or not the determined value of the correction ζ_(P)^(Δ) satisfies Equation 52 may be repeated iteratively, e.g., untilEquation 52 is satisfied. The value of d₁ may be selected according toany suitable solver algorithm.

In some demonstrative embodiments, the method of determining the valueof d₁ may be performed using any suitable solver, for example, a solverincluding bisection for convergence and/or stability. In one embodiment,the solver may include a Newton-Raphson solver. In other embodiments,the solver may include any other suitable solver type, e.g., a Brentsolver and the like.

In some demonstrative embodiments, Equations 17 and/or 18 may besimplified using any suitable approximation, e.g., in order to allowsolving of Equations 23 and/or 24 in a more efficient and/or quickermanner. In one example, Equations 17 and/or 18 may be rewritten usingthe format of a Taylor-series approximation, e.g., as follows:

$\begin{matrix}{{{\zeta_{C}\left( {\sigma_{K_{Call}},K_{Call}} \right)} = {\left( {\sigma_{K_{Call}} - \sigma_{0}} \right){{df}_{L} \cdot S}{\sqrt{t} \cdot {N\left( d_{1} \right)}}}}{\cdot \left( {1 + {\left( {\sigma_{K_{Call}} - \sigma_{0}} \right)\frac{d_{1}}{2}\left( {\frac{d_{1}}{\sigma_{K_{Call}}} - \sqrt{t}} \right)}} \right)}} & (55) \\{{{\zeta_{P}\left( {\sigma_{K_{Put}},K_{Put}} \right)} = {\left( {\sigma_{K_{Put}} - \sigma_{0}} \right){{df}_{L} \cdot S}{\sqrt{t} \cdot {N\left( d_{1} \right)}}}}{\cdot \left( {1 + {\left( {\sigma_{K_{Put}} - \sigma_{0}} \right)\frac{d_{1}}{2}\left( {\frac{d_{1}}{\sigma_{K_{Put}}} - \sqrt{t}} \right)}} \right)}} & (56)\end{matrix}$

In some demonstrative embodiments, pricing module 160 may receive fromuser 102, e.g., via interface 110, first input data including one ormore parameters defining an option to be priced (“the requestedoption”).

In some demonstrative embodiments, pricing module 160 may receive, e.g.,from market data service 149, second input data corresponding to atleast one current market condition relating to an underlying asset ofthe option, e.g., including real time market data corresponding to anasset class of the requested option.

For example, for FX instruments, pricing module 160 may receive frommarket data service 149 market data including one or more of spot rates,forward rates, interest rates, at the money volatility for differentmaturities, 25 delta risk reversals for different maturities, 25 deltabutterflies for different maturities and, optionally, other delta riskreversals and/or butterflies, e.g., the 10 delta risk reversal and/orthe 10 delta butterfly.

For interest-rate instruments, pricing module 160 may receive frommarket data service 149 market data including one or more of Liborrates, e.g., all Libor rates in all available countries, swap rates forall maturities, interest-rates future prices in currencies, whereavailable, cap floor volatilities or prices for several strikes,swaption at the money volatilities and other strikes such as 100 or 200basis points over and under the at the money forward strike.

For equity options pricing module 160 may receive from market dataservice 149 market data including exchange prices for stocks andindices, exchange prices for options on stocks and indices, forwardprices for several maturities, and/or security lending rates andinterest rates, and the like.

In some demonstrative embodiments, pricing module 160 may determine thefunctions A(Δ) and/or B(Δ), for example, based on the received marketdata, e.g., using Equations 31-34, as described above.

In some demonstrative embodiments, pricing module 160 may determine thevolatility smile corresponding to the option. For example, pricingmodule may determine one or more volatilities σ_(k) corresponding to aVanilla option having one or more respective strikes K, for example,based on Equations 23 and/or 27, e.g., depending on the whether theoption is a call option or a put option.

In some demonstrative embodiments, pricing module 160 may perform anysuitable extrapolation and/or interpolation operations to determine avolatility surface and/or the volatility corresponding to the strike andexpiration time of the requested option, e.g., based on the determinedvolatilities σ_(k).

In some demonstrative embodiments, pricing module 160 may determine thecorrection ζ to be added to the BS value of the Vanilla option inaccordance with the volatility smile, for example, according toEquations 23 and/or 27, e.g., depending on the whether the option is acall option or a put option.

In some demonstrative embodiments, pricing module 160 may determine theprice of the Vanilla option based on the correction ζ and the BS valueof the Vanilla option, e.g., according to Equation 16.

In some demonstrative embodiments, pricing module 160 may determine theprice of the requested option, e.g., based on the determined price ofthe corresponding Vanilla option.

In some demonstrative embodiments, interface 110 and pricing module 160may be implemented as part of an application or application server toprocess user information, e.g., including details of a defined option tobe priced, received from user 102, as well as real time tradeinformation, received, for example, from market data service 149. System100 may also include storage 161, e.g., a database, for storing the userinformation and/or the trade information.

The user information may be received from user 102, for example, via acommunication network, for example, the Internet, e.g., using a directtelephone connection or a Secure Socket Layer (SSL) connection, a LocalArea Network (LAN), or via any other communication network known in theart. Pricing module 160 may communicate a determined price correspondingto the defined option to user 102 via interface 110, e.g., in a formatconvenient for presentation to user 102.

A system, e.g., system 100, for pricing financial derivatives accordingto some embodiments, may provide price information for substantially anysuitable option on substantially any suitable asset based on inputmarket data. The market data may be easily obtained, e.g., by pricingmodule 160, on a real time basis. Thus, pricing module 160 may provideuser 102 with a real-time price of any desired option, e.g., based onreal time prices received from the exchanges and/or OTC market. Pricingmodule 160 may update the price, e.g., substantially immediately and/orautomatically, for example, in response to a change in spot pricesand/or option prices. This may enable user 102 to automatically updateprices for trading with the exchanges.

A trader may want, for example, to submit a plurality of bid and/oroffer (hereinafter “bid/offer”) prices for a plurality of options, e.g.,ten bid/offer prices for ten options, respectively. When entering thebids/offers to a quoting system, the trader may check the price, e.g.,in relation to the current spot prices, and may then submit thebids/offers to the exchange. Some time later, e.g., a second later, thespot price of the stock which is the underlying asset of one or more ofthe options may change. A change in the spot prices may be accompanied,for example, by changes in the volatility parameters, or may includejust a small spot change while the volatility parameters have notchanged. In response to the change in the spot price, the trader maywant to update one or more of the submitted bid/offer prices. The desireto update the bid/offer prices may occur, e.g., frequently, during tradetime.

A system according to some demonstrative embodiments, e.g., system 100,may automatically update the bid/offer prices entered by the trader,e.g., based on any desired criteria. For example, pricing module 160 mayevaluate the trader's bids/offers versus bid and offer prices of theoptions, which may be estimated by pricing module 160, e.g., when thetrader submits the bid/offer prices. Pricing module 160 may thenautomatically recalculate the bid and/or offer prices, e.g., wheneverthe spot changes, and may automatically update the trader's bid/offerprices. Pricing module 160 may, for example, update one or more of thetrader's bid/offer prices such that a price difference between thebid/offer price calculated by pricing module 160 and the trader'sbid/offer price is kept substantially constant. According to anotherexample, pricing module 160 may update one or more of the trader'sbid/offer prices based on a difference between the trader's bid/offerprices and an average of bid and offer prices calculated by pricingmodule 160. Pricing module 160 may update one or more of the trader'sbid/offer prices based on any other desired criteria.

It is noted, that a change of the spot price, e.g., of a few pips, mayresult in a change in one or more of the volatility parameters ofoptions corresponding to the spot price. It will be appreciated that apricing module according to some embodiments, e.g., pricing module 160,may enable automatically updating one or more option prices submitted bya trader, e.g., while taking into account the change in the spot price,in one or more of the volatility parameters, and/or in any other desiredparameters, as described above.

According to some demonstrative embodiments, pricing module 160 mayenable the trader to submit one or more quotes in the exchange in a formof relative prices vs. prices determined by the pricing module 160. Forexample, the trader may submit quotes for one or more desired strikesand/or expiry dates. The quotes submitted by the trader may be in anydesired form, e.g., relating to one or more corresponding pricesdetermined by pricing module 160. For example, the quotes submitted bythe trader may be in the terms of the bid/offer prices determined bypricing module 160 plus two basis points; in the terms of the mid marketprice determined by pricing module 160 minus four basis points, and/orin any other suitable format and/or terms. Pricing module 160 maydetermine the desired prices, for example, in real time, e.g., whenevera price change in the exchange is recorded. Alternatively, pricingmodule 160 may determine the desired prices, according to any otherdesired timing scheme, for example, every predefined time interval,e.g., every half a second.

A change in a spot price of a stock may result in changes in the pricesof a large number of options related to the stock. For example therecould be over 200 active options relating to a single stock and havingdifferent strikes and expiration dates. Accordingly, a massive bandwidthmay be required by traders for updating the exchange prices of theoptions in accordance with the spot price changes, e.g., in real time.This may lead the traders to submit to the exchange prices which may be“non-competitive”, e.g., prices including a “safety-margin”, since thetraders may not be able to update the submitted prices according to therate at which the spot prices, the volatility, the dividend, and/or thecarry rate may change.

According to some demonstrative embodiments, pricing module 160 may beimplemented, e.g., by the exchange or by traders, for example, toautomatically update one or more bid and/or offer prices submitted by atrader, e.g., as described above. This may encourage the traders tosubmit with the exchange more aggressive bid and/or offer prices, sincethe traders may no longer need to add the “safety margin” their pricesfor protecting the traders against the frequent changes in the spotprices. Accordingly, the trading in the exchange may be more effective,resulting in a larger number of transactions. For example, a trader mayprovide pricing system 100 with one or more desired volatility parameterand/or rates. The trader may request system 100 to automatically submitand/or update bid and/or offer prices on desired amounts of options,e.g., whenever there is a significant change in the spot price and/or inthe volatility of the market. The trader may also update some or all ofthe volatility parameters. In addition, system 100 may be linked, forexample, to an automatic decision making system, which may be able todecide when to buy and/or sell options using pricing module 160.

Reference is made to FIG. 2, which schematically illustrates a method ofpricing an option in accordance with some demonstrative embodiments. Insome demonstrative embodiments, one or more of the operations of themethod of FIG. 2 may be performed and/or implemented by any suitabledevice and/or system, for example, suitable computing device and/orsystem, e.g., system 100 (FIG. 1) and/or pricing module 160 (FIG. 1).

As indicated at block 202, the method may include receiving first inputdata corresponding to at least one parameter defining a first option onan underlying asset. For example, module 160 (FIG. 1) may receive e.g.,from user 102 (FIG. 1), the first input data defining an option to bepriced, e.g., as described above.

As indicated at block 204, the method may include receiving second inputdata corresponding to at least one current market condition relating tothe underlying asset. For example, module 160 (FIG. 1) may receive e.g.,from services 149 (FIG. 1), the second input data corresponding to theunderlying asset, e.g., as described above.

As indicated at block 206, the method may include determining a price ofthe first option based on the first and second input data, according toa volatility smile satisfying one or more predefined criterions.

As indicated at block 208, determining the price of the first option mayinclude determining the price of the first option according to avolatility smile satisfying a first criterion relating to a sum of afirst correction corresponding to the first option and a secondcorrection corresponding to a second option representing a positionopposite to a position of a the first option and having a same delta asthe first option.

In some demonstrative embodiments, the first correction may relate to adifference between a theoretical price of the first option and the priceof the first option according to the volatility smile, and/or the secondcorrection may relate to a difference between a theoretical price of thesecond option and the price of the second option according to thevolatility smile. For example, module 160 (FIG. 1) may determine theprice of the first option according to a volatility smile satisfyingEquations 19 and 20, e.g., as described above.

As indicated at block 210, determining the price of the first optionaccording to the volatility smile may include determining market-basedparameters of first and second proportionality functions based on thesecond input data. For example, module 160 (FIG. 1) may determine themarket-based parameters of the proportionality functions A(Δ) and B(Δ)based on the market data, e.g., as described above.

As indicated at block 212, determining the price of the first optionaccording to the volatility smile may include determining the firstcorrection based on the first and second criterions. For example, module160 (FIG. 1) may determine the correction corresponding to the firstoption according to Equations 23 and/or 27, e.g., as described above.

As indicated at block 214 determining the first correction may includedetermining a volatility of the first option based on the first andsecond criterions, and determining the first correction based on thevolatility of the first option. For example, module 160 (FIG. 1) maydetermine the volatility σ corresponding to the first option, and thecorrection ζ corresponding to the volatility σ, e.g., as describedabove.

Following are examples of volatility smiles determined with respect tooptions on various asset classes, using the volatility smile mode asdescribed herein in accordance with some demonstrative embodiments. Itshould be noted that the trade information used in these examples havebeen randomly selected from the market for demonstrative purposes onlyand is not intended to limit the scope of the embodiments describedherein to any particular choice of the trade information.

The volatility smiles were determined using the followingproportionality functions:A(Δ)=c _(1e) ^(−C) ² ^((Δ) ⁰ ^(−Δ))  (57)B(Δ)=c′ _(1e) ^(−c′) ² ^((Δ) ⁰ ^(−Δ))  (58)wherein c₁, c₁′, c₂, c₂′ denote four respective market parameters to bedetermined, e.g., based on the traded market data.

The following examples demonstrate the results of the volatility smilemodel with respect to different asset classes, e.g., at the same time.The following examples relate to options on currencies, e.g., options onthe exchange rate of EURO (EUR) to US dollar (USD) (EUR/USD), which aretraded in the OTC market; options on Interest Rates, e.g., swaptions onEUR swap rates, which are traded in the OTC market; options onCommodities, e.g., options on West Texas intermediate (WTI) crude oil,which is exchange traded; and options on Equities, e.g., options on theDAX index, which is exchange traded. All of the examples relate toassets, which are very liquid and commonly traded, therefore the marketdata may be assumed to be accurate. The examples relate to differentmaturities. The following examples are based on market data on Dec. 27,2010.

A first example relates to FX options on EUR/USD with an expiration ofone year. The FX options market trades ATM delta neutral volatility aswell as delta strikes. The inputs received from the market aresummarized in Table 1:

TABLE 1 Delta neutral ATM vol σ₀ = 14.45; Forward rate = 1.31408 Delta 5Δ 10 Δ 25 Δ 25 Δ 10 Δ 5 Δ Put Put Put ATM Call Call Call Strike 0.9511.053 1.1956 1.3279 1.4541 1.5933 1.7016 Market Vol 21.19 18.775 16.22514.45 13.825 14.325 15.08

Based on the above market data, the market-based parameters may bedetermined as follows, e.g., using the model described above: c1=0.002,c2=0.5, c1′=0.0042, c2′=1.6.

A volatility smile (“the model volatility smile”) corresponding to theFX options may be determined according to Equations 19 and 20, e.g., asdescribed above. Table 2 includes seven volatilities corresponding toseven respective strikes determined according to the volatility smile:

TABLE 2 Strike 0.9423 1.0562 1.1941 1.3279 1.4541 1.5933 1.7016 ModelVol 21.638 18.533 16.288 14.450 13.723 14.343 15.023

FIG. 3A schematically illustrates a first graph 302 depicting the modelvolatility smile based on Table 2, and a second graph 304 depicting themarket volatilities of Table 1. As shown in FIG. 3A, the differencesbetween the model volatility smile and the market volatilities aregenerally negligible.

A second example relates to options on EUR swaps rate with a maturity often years and an expiration of one year. The interest-rates markettrades ATM forward strikes (ATMF, where the strike is the forward rate)and other strikes may be measured with respect to a difference in basispoints from the forward rate. The inputs received from the market aresummarized in Table 3:

TABLE 3 Forward = 3.671 Market Data −100 −50 −25 ATMF +25 +50 +100 +200Strike 2.671 3.171 3.421 3.671 3.921 4.171 4.671 5.671 Market Vol 31.327.7 26.3 25.1 24 23.2 22.2 21.9

Based on the above market data, the market-based parameters may bedetermined as follows, e.g., using the model described above: σ₀=24.5,c1=0.0045, c2=1.5, c1′=0.0095, c2′=0.1.

The model volatility smile corresponding to the IR options may bedetermined according to Equations 19 and 20, e.g., as described above.Table 4 includes volatilities corresponding to respective strikesdetermined according to the volatility smile:

TABLE 4 Strike 2.644177 2.874236 3.12931 3.387316 3.643633 3.921 ModelVol 31.21694 29.41119 27.83829 26.42833 25.13876 23.94235 Strike 4.1714.421 4.671 4.9 5.671 Model Vol 23.12829 22.52502 22.11125 21.8875821.9418

FIG. 3B schematically illustrates a first graph 306 depicting the modelvolatility smile based on Table 4 and a second graph 308 depicting themarket volatilities of Table 3. As shown in FIG. 3B, the differencesbetween the model volatility smile and the market volatilities aregenerally negligible.

A third example relates to options on WTI crude oil with expiration onNov. 15, 2012 (687 days). The underlying asset of these options is theWTI future contract of December 12 (December 2012). The market data istaken from the Nymex exchange (CME), and includes about 20 strikes withtheir corresponding volatility implied from the exchange price foroption premium. The inputs received from the market are summarized inTable 5:

TABLE 5 Forward = 92.61 Strike 65 70 75 80 85 90 95 100 105 110 MarketVol 29.75 28.97 28.06 27.04 26.32 25.68 25.07 24.47 23.89 23.8 Strike115 120 125 130 135 140 145 150 160 Market Vol 23.65 23.65 23.74 23.8324.3 24.16 24.4 24.6 25.02

Based on the above market data, the market-based parameters may bedetermined as follows, e.g., using the model described above: σ₀=24.491,c1=0.0105, c2=0.015, c1′=0.0165, c2′=0.65.

The model volatility smile corresponding to the WTI options may bedetermined according to Equations 19 and 20, e.g., as described above.Table 6 includes volatilities corresponding to respective strikesdetermined according to the volatility smile:

TABLE 6 Strike 64.52 70.87 78.17 82.23 86.58 91.21 96.05 100.00 105.00110.00 Model Vol 30.20 28.76 27.38 26.71 26.05 25.40 24.79 24.35 23.9423.69 Strike 115 120 125 130 135 140 145 150 160 Model Vol 23.58 23.5823.66 23.80 23.98 24.19 24.41 24.65 25.15

FIG. 3C schematically illustrates a first graph 310 depicting the modelvolatility smile based on Table 6 and a second graph 312 depicting themarket volatilities of Table 5. As shown in FIG. 3C, the differencesbetween the model volatility smile and the market volatilities aregenerally negligible.

A fourth example relates to options on the DAX index with expiration onDec. 21, 2012 (725 calendar days). The market volatilities are takenfrom the exchange settlement prices for the expiry date of Dec. 21,2012. The inputs received from the market are summarized in Table 7:

TABLE 7 Forward = 7187.635 Strike 4200 4600 5000 5400 5800 6200 66007000 7400 Market Vol 33.16 31.62 30.14 28.72 27.33 25.95 24.60 23.3022.11 Strike 7600 8000 8400 8800 9200 9600 10000 10400 11000 Market Vol21.57 20.61 19.82 19.19 18.65 18.16 17.72 17.32 17.02

Based on the above market data, the market-based parameters may bedetermined as follows, e.g., using the model described above: σ₀=22.00,c1=0.005, c2=0.2, c1′=0.025, c2′=0.1.

The model volatility smile corresponding to the DAX options may bedetermined according to Equations 19 and 20, e.g., as described above.Table 8 includes volatilities corresponding to respective strikesdetermined according to the volatility smile.

TABLE 8 Strike 4238.85 4693.35 5159.1 5639.83 6127.86 6606.64 7060.757482.51 7600 Model Vol 33.79 31.31 29.26 27.50 25.94 24.54 23.29 22.1521.84 Strike 8000 8400 8800 9200 9600 10000 10400 11000 Model Vol 20.8119.87 19.06 18.45 18.05 17.85 17.80 17.89

FIG. 3D schematically illustrates a first graph 314 depicting the modelvolatility smile based on Table 8 and a second graph 316 depicting themarket volatilities of Table 7. As shown in FIG. 3D, the differencesbetween the model volatility smile and the market volatilities aregenerally negligible.

Reference is made to FIG. 4, which schematically illustrates an articleof manufacture 400, in accordance with some demonstrative embodiments.Article 400 may include a machine-readable storage medium 402 to storelogic 404, which may be used, for example, to perform at least part ofthe functionality of pricing module 160 (FIG. 1); and/or to perform oneor more operations described herein.

In some demonstrative embodiments, article 400 and/or machine-readablestorage medium 402 may include one or more types of computer-readablestorage media capable of storing data, including volatile memory,non-volatile memory, removable or non-removable memory, erasable ornon-erasable memory, writeable or re-writeable memory, and the like. Forexample, machine-readable storage medium 402 may include, RAM, DRAM,Double-Data-Rate DRAM (DDR-DRAM), SDRAM, static RAM (SRAM), ROM,programmable ROM (PROM), erasable programmable ROM (EPROM), electricallyerasable programmable ROM (EEPROM), Compact Disk ROM (CD-ROM), CompactDisk Recordable (CD-R), Compact Disk Rewriteable (CD-RW), flash memory(e.g., NOR or NAND flash memory), content addressable memory (CAM),polymer memory, phase-change memory, ferroelectric memory,silicon-oxide-nitride-oxide-silicon (SONOS) memory, a disk, a floppydisk, a hard drive, an optical disk, a magnetic disk, a card, a magneticcard, an optical card, a tape, a cassette, and the like. Thecomputer-readable storage media may include any suitable media involvedwith downloading or transferring a computer program from a remotecomputer to a requesting computer carried by data signals embodied in acarrier wave or other propagation medium through a communication link,e.g., a modem, radio or network connection.

In some demonstrative embodiments, logic 404 may include instructions,data, and/or code, which, if executed by a machine, may cause themachine to perform a method, process and/or operations as describedherein. The machine may include, for example, any suitable processingplatform, computing platform, computing device, processing device,computing system, processing system, computer, processor, or the like,and may be implemented using any suitable combination of hardware,software, firmware, and the like.

In some demonstrative embodiments, logic 404 may include, or may beimplemented as, software, a software module, an application, a program,a subroutine, instructions, an instruction set, computing code, words,values, symbols, and the like. The instructions may include any suitabletype of code, such as source code, compiled code, interpreted code,executable code, static code, dynamic code, and the like. Theinstructions may be implemented according to a predefined computerlanguage, manner or syntax, for instructing a processor to perform acertain function. The instructions may be implemented using any suitablehigh-level, low-level, object-oriented, visual, compiled and/orinterpreted programming language, such as C, C++, Java, BASIC, Matlab,Pascal, Visual BASIC, assembly language, machine code, and the like.

The processes and displays presented herein are not inherently relatedto any particular computer or other apparatus. Various general-purposesystems may be used with programs in accordance with the teachingsherein, or it may prove convenient to construct a more specializedapparatus to perform the desired method. The desired structure for avariety of these systems will appear from the description below. Inaddition, some embodiments are not described with reference to anyparticular programming language. It will be appreciated that a varietyof programming languages may be used to implement the teachings of theinvention as described herein.

Functions, operations, components and/or features described herein withreference to one or more embodiments, may be combined with, or may beutilized in combination with, one or more other functions, operations,components and/or features described herein with reference to one ormore other embodiments, or vice versa.

While certain features of the invention have been illustrated anddescribed herein, many modifications, substitutions, changes, andequivalents may occur to those skilled in the art. It is, therefore, tobe understood that the appended claims are intended to cover all suchmodifications and changes as fall within the true spirit of theinvention.

What is claimed is:
 1. A system of pricing options, the systemcomprising: a memory having stored thereon pricing-module instructions;and a processor to execute the pricing-module instructions resulting ina pricing module configured to: receive first input data correspondingto at least one parameter defining a first option on an underlyingasset; receive second input data corresponding to at least one currentmarket condition relating to said underlying asset; and determine, basedon said first and second input data, a price of the first optionaccording to a volatility smile, wherein the volatility smile is definedas satisfying a first criterion and a second criterion, the firstcriterion relates to a sum of a first correction corresponding to thefirst option and a second correction corresponding to a second option,the second criterion relates to a difference between the firstcorrection corresponding to the first option and the second correctioncorresponding to the second option, wherein the second option representsa position opposite to a position of the first option and has a sameabsolute delta value as the first option, wherein the first correctionrelates to a difference between the price of the first option accordingto the volatility smile and a theoretical price of the first option, andwherein the second correction relates to a difference between the priceof the second option according to the volatility smile and a theoreticalprice of the second option.
 2. The system of claim 1, wherein the firstcriterion requires that the sum of the first and second corrections isproportional to a sum of first and second volatility convexitiescorresponding to the first and second options, respectively, and whereinthe second criterion requires that a difference between the first andsecond corrections is proportional to a difference between first andsecond delta convexities corresponding to the first and second options,respectively.
 3. The system of claim 2, wherein the sum of the first andsecond volatility convexities is a predefined function of a volatilityof the first option according to the volatility smile and a volatilityof the second option according to the volatility smile, and wherein thedifference between the first and second delta convexities is a secondpredefined function of the volatility of the first option according tothe volatility smile and the volatility of the second option accordingto the volatility smile.
 4. The system of claim 3, wherein the firstcriterion requires that the sum of the first and second corrections isproportional to the sum of first and second volatility convexitiesaccording to a first proportionality function, which is based on saiddelta, and wherein the second criterion requires that the differencebetween the first and second corrections is proportional to thedifference between the first and second delta convexities according to asecond proportionality function, which is based on said delta.
 5. Thesystem of claim 4, wherein at least one of the first and secondproportionality functions includes a predefined combination of saiddelta and one or more market-based parameters, and wherein said pricingmodule is to determine said market-based parameters based on said secondinput data.
 6. The system of claim 4, wherein the first and secondproportionality functions are decreasing functions of said delta.
 7. Thesystem of claim 2, wherein the first and second criterion requiresatisfying the following equations, respectively:${\zeta_{C}^{\Delta} + \zeta_{P}^{\Delta}} = {{{A(\Delta)} \cdot {Vega}^{\Delta}}{d_{1}^{2}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}}$${\zeta_{C}^{\Delta} - \zeta_{P}^{\Delta}} = {{{B(\Delta)} \cdot {Vega}^{\Delta}}\frac{d_{1}}{S\sqrt{t}}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}$wherein ζ_(C) ^(Δ) and ζ_(P) ^(Δ) denote said first and secondcorrections, wherein Δ denotes said delta, wherein A(Δ) and B(Δ) denotefirst and second functions of Δ, respectively, wherein Vega^(Δ) denotesa vega of the first and second options, wherein t denotes a time toexpiration of said first option, wherein d₁ denotes a predefinedfunction of the time to expiration of said first option, wherein Sdenotes a price of said underlying asset, and wherein σ_(K) _(Call) andσ_(K) _(Put) denote a volatility of the first option according to thevolatility smile and a volatility of the second option according to thevolatility smile, respectively.
 8. The system of claim 1, wherein saidpricing module is to determine said first correction based on the firstand second criterions, and to determine the price of said first optionbased on the first correction and said theoretical price of the firstoption.
 9. The system of claim 8, wherein said pricing module is todetermine a volatility of the first option based on the first and secondcriterions, and to determine said first correction based on thevolatility of the first option.
 10. The system of claim 1, wherein saidfirst and second options include Vanilla options.
 11. The system ofclaim 10, wherein said pricing module is to determine a price of anexotic option on said underlying asset based on the volatility smile.12. The system of claim 1, wherein said first input data comprises anindication of at least one parameter selected from the group consistingof a type of said first option, an expiration date of said first option,a trigger for said first option, and a strike of said first option. 13.The system of claim 1, wherein said second input data comprises anindication of at least one parameter selected from the group consistingof a spot value, a forward rate, an interest rate, a volatility, anat-the-money volatility, a delta risk reversal, a delta butterfly, adelta strangle, a 10 delta risk reversal, a 10 delta butterfly, a 10delta strangle, a 25 delta risk reversal, a 25 delta butterfly, a 25delta strangle, a caplet, a floorlet, a swap rate, a security lendingrate, and an exchange price.
 14. The system of claim 1, wherein saidpricing module is to provide an output based on the price of the firstoption.
 15. The system of claim 14, wherein said pricing module is tocommunicate said output via a communication network.
 16. The system ofclaim 1, wherein said underlying asset comprises a financial asset. 17.The system of claim 1, wherein said underlying asset is related to atleast one asset type selected from the group consisting of a commodity,a stock, a bond, a currency, an interest rate, and the weather.
 18. Acomputer-based method of pricing options, the method comprising:receiving, by a computing device, first input data corresponding to atleast one parameter defining an option to be priced on an underlyingasset; receiving, by the computing device, second input datacorresponding to at least one current market condition relating to saidunderlying asset; and determining by the computing device, based on saidfirst and second input data, a price of the option according to avolatility smile, wherein the volatility smile is defined as satisfyinga first criterion and a second criterion, the first criterion relates toa sum of a first correction corresponding to a first option and a secondcorrection corresponding to a second option, the second criterionrelates to a difference between the first correction corresponding tothe first option and the second correction corresponding to the secondoption, wherein the second option represents a position opposite to aposition of the first option and has a same absolute delta value as thefirst option, wherein the first correction relates to a differencebetween the price of the first option according to the volatility smileand a theoretical price of the first option, and wherein the secondcorrection relates to a difference between the price of the secondoption according to the volatility smile and a theoretical price of thesecond option.
 19. The method of claim 18, wherein the first criterionrequires that the sum of the first and second corrections isproportional to a sum of first and second volatility convexitiescorresponding to the first and second options, respectively, and whereinthe second criterion requires that the difference between the first andsecond corrections is proportional to a difference between first andsecond delta convexities corresponding to the first and second options,respectively.
 20. The method of claim 19, wherein the sum of the firstand second volatility convexities is a predefined function of avolatility of the first option according to the volatility smile and avolatility of the second option according to the volatility smile, andwherein the difference between the first and second delta convexities isa second predefined function of the volatility of the first optionaccording to the volatility smile and the volatility of the secondoption according to the volatility smile.
 21. The method of claim 20,wherein the first criterion requires that the sum of the first andsecond corrections is proportional to the sum of first and secondvolatility convexities according to a first proportionality function,which is based on said delta, and wherein the second criterion requiresthat the difference between the first and second corrections isproportional to the difference between the first and second deltaconvexities according to a second proportionality function, which isbased on said delta.
 22. The method of claim 21, wherein at least one ofthe first and second proportionality functions includes a predefinedcombination of said delta and one or more market-based parameters, themethod including determining said market-based parameters based on saidsecond input data.
 23. The method of claim 21, wherein the first andsecond proportionality functions are decreasing functions of said delta.24. The method of claim 19, wherein the first and second criterionrequire satisfying the following equations, respectively:${\zeta_{C}^{\Delta} + \zeta_{P}^{\Delta}} = {{{A(\Delta)} \cdot {Vega}^{\Delta}}{d_{1}^{2}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}}$${\zeta_{C}^{\Delta} - \zeta_{P}^{\Delta}} = {{{B(\Delta)} \cdot {Vega}^{\Delta}}\frac{d_{1}}{S\sqrt{t}}\left( {\frac{1}{\sigma_{K_{Call}}} + \frac{1}{\sigma_{K_{Put}}}} \right)}$wherein ζ_(C) ^(Δ) and ζ_(P) ^(Δ) denote said first and secondcorrections, wherein Δ denotes said delta, wherein A(Δ) and B(Δ) denotefirst and second functions of Δ, respectively, wherein Vega^(Δ) denotesa vega of the first and second options, wherein t denotes a time toexpiration of said first option, wherein d₁ denotes a predefinedfunction of the time to expiration of said first option, wherein Sdenotes a price of said underlying asset, and wherein σ_(K) _(Call) andσ_(K) _(Put) denote a volatility of the first option according to thevolatility smile and a volatility of the second option according to thevolatility smile, respectively.
 25. The method of claim 18, wherein saidfirst and second options include Vanilla options.
 26. The method ofclaim 25 including determining a price of an exotic option on saidunderlying asset based on the volatility smile.
 27. The method of claim18, wherein said first input data comprises an indication of at leastone parameter selected from the group consisting of a type of said firstoption, an expiration date of said first option, a trigger for saidfirst option, and a strike of said first option.
 28. The method of claim18, wherein said second input data comprises an indication of at leastone parameter selected from the group consisting of a spot value, aforward rate, an interest rate, a volatility, an at-the-moneyvolatility, a delta risk reversal, a delta butterfly, a delta strangle,a 10 delta risk reversal, a 10 delta butterfly, a 10 delta strangle, a25 delta risk reversal, a 25 delta butterfly, a 25 delta strangle, acaplet, a floorlet, a swap rate, a security lending rate, and anexchange price.
 29. The method of claim 18 including providing an outputbased on the price of the first option.
 30. The method of claim 18,wherein said underlying asset comprises a financial asset.
 31. Themethod of claim 18, wherein said underlying asset is related to at leastone asset type selected from the group consisting of a commodity, astock, a bond, a currency, an interest rate, and the weather.